Apparatus for and method of measuring a jitter

ABSTRACT

There is provided an apparatus for and a method of measuring a jitter wherein a clock waveform X C (t) is transformed into an analytic signal using Hilbert transform and a varying term Δφ(t) of an instantaneous phase of this analytic signal is estimated.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus for and a method ofmeasuring a jitter in a microcomputer. More particularly, the presentinvention relates to an apparatus for and a method of measuring a jitterin a clock generating circuit used in a microcomputer.

2. Description of the Related Art

In the past thirty years, the number of transistors on a VLSI (verylarge scale integrated circuit) chip has been exponentially increasingin accordance with Moore's law, and the clock frequency of amicrocomputer has also been exponentially increasing in accordance withMoore's law. At present time, the clock frequency is about to exceed thelimit of 1.0 GHz. (For example, see: Naoaki Aoki, H. P. Hofstee, and S.Dong; “GHz MICROPROCESSOR”, INFORMATION PROCESSING vol. 39, No. 7, July1998.) FIG. 1 is a graph showing a progress of clock period in amicrocomputer disclosed in Semiconductor Industry Association: “TheNational Technology Roadmap for Semiconductors, 1997”. In FIG. 1, an RMSjitter (root mean square jitter) is also plotted.

In a communication system, a carrier frequency and a carrier phase, or asymbol timing are regenerated by applying non-linear operations to areceived signal and by inputting the result of the non-linear process toa phase-locked loop (PLL) circuit. This regeneration corresponds to themaximum likelihood parameter estimation. However, when a carrier or adata cannot correctly be regenerated from the received signal due to aninfluence of a noise or the like, a retransmission can be requested tothe transmitter. In a communication system, a clock generator is formedon a separate chip from the other components. This clock generator isformed on a VLSI chip using a bipolar technology, GaAs technology or aCMOS technology.

In each of many microcomputers, an instruction execution is controlledby a clock signal having a constant period. The clock period of thisclock signal corresponds to a cycle time of a microcomputer. (Forexample, see: Mike Johnson; “Superscale Microprocessor Design”,Prentice-Hall, Inc., 1991.) If the clock period is too short, asynchronous operation becomes impossible and the system is locked. In amicrocomputer, a clock generator is integrated in a same chip whereother logical circuits are integrated. FIG. 2 shows, as an example, aPentium chip. In FIG. 2, a white square (□) indicates a clock generatingcircuit. This microcomputer is produced utilizing a CMOS (complementarymetal-oxide semiconductor) processing.

In a communication system, the average jitter or the RMS jitter isimportant. The RMS jitter contributes to an average noise ofsignal-to-noise ratio and increases the bit error rate. On the otherhand, in a microcomputer, the worst instantaneous value of someparameter determines the operation frequency. That is, the peak-to-peakjitter (the worst value of jitter) determines the upper limit of theoperation frequency.

Therefore, for testing of a PLL circuit in a microcomputer, there isrequired a method of measuring an instantaneous value of jitteraccurately and in a short period of time. However, since a measurementof a jitter has been developed in the area of communications, there isno measuring method, in the present state, corresponding to thisrequirement in the area of microcomputers. It is an object of thepresent invention to provide a method of measuring an instantaneousvalue of jitter accurately and in a short period of time.

On the contrary, for testing of a PLL circuit in a communication system,there is required a method of measuring an RMS jitter accurately.Although it takes approximately 10 minutes of measuring time, ameasuring method actually exists and is practically used. FIG. 3collectively shows comparisons of clock generators between amicrocomputer and a communication system.

A phase-locked loop circuit (PLL circuit) is a feedback system. In a PLLcircuit, a frequency and a phase θ_(i) of a given reference signal arecompared with a frequency and a phase θ₀ of an internal signal source,respectively to control the internal signal source, using thedifferences therebetween, such that the frequency difference or thephase difference can be minimized. Therefore, a voltage controlledoscillator (VCO) which is an internal signal source of a PLL circuitcomprises a component or components the delay time of which can bevaried. When a DC voltage is inputted to this oscillator, a repetitivewaveform having a constant period proportional to the direct currentvalue is outputted.

The PLL circuit relating to the present invention comprises aphase-frequency detector, a charge pump circuit, a loop filter and aVCO. FIG. 4 shows a basic circuit configuration of a PLL circuit in ablock diagram form. Next, the operation of each of the circuitcomponents will be briefly described.

A phase-frequency detector is a digital sequential circuit. FIG. 5 is ablock diagram showing a circuit configuration of a phase-frequencydetector comprising two D-type flip-flops D-FF1 and D-FF2 and an ANDgate. A reference clock is applied to a clock terminal ck of the firstD-type flip-flop D-FF1, and a PLL clock is applied to a clock terminalck of the second D-type flip-flop D-FF2. A logical value “1” is suppliedto each data input terminal D.

In the circuit configuration described above, when each of the two Qoutputs of the both flip-flops becomes “1” at the same time, the ANDgate resets the both flip-flops. The phase-frequency detector outputs,depending on the phase difference and the frequency difference betweenthe two input signals, an UP signal for increasing the frequency or aDOWN signal for decreasing the frequency. (For example, see: R. JacobBaker, Henry W. Li, and David E. Boyce; “CMOS Circuit Design, Layout,and Simulation”, IEEE Press, 1998.)

FIG. 6 shows a state transition diagram of a phase-frequency detector(PFD). The phase-frequency detector transits the state by rise edges ofa reference clock and a PLL clock. For example, as shown in FIG. 7, whenthe frequency of a reference clock is 40 MHz and the frequency of a PLLclock is 37 MHz, in order to increase the frequency, an UP signal isoutputted during a time interval between the two rise edges. A similaroperation is also performed when a phase difference is present betweenthe reference clock and the PLL clock. The phase-frequency detector hasthe following characteristics compared with a phase detector using anExclusive OR circuit. (For example, see: R. Jacob Baker, Henry W. Li,and David E. Boyce; “CMOS Circuit Design, Layout, and Simulation”, IEEEPress, 1998.)

(i) The phase-frequency detector operates at a rising edge of an inputclock, and does not relate to the shape of the waveform such as a pulsewidth of the clock.

(ii) The phase-frequency detector is not locked by a harmonic of thereference frequency.

(iii) Since both of the two outputs are logical “0” during a time periodwhen the loop is locked, a ripple is not generated at the output of theloop filter.

The phase-frequency detector is highly sensitive to an edge. When anedge of a reference clock cannot be discriminated due to a noise, thephase-frequency detector is hung-up to some state. On the other hand, ina phase detector based on an Exclusive OR circuit, even if an edgecannot be discriminated, the average output is 0 (zero). Therefore,

(iv) the phase-frequency detector is sensitive to a noise.

A charge pump circuit converts logical signals UP and DOWN from thephase-frequency detector (PFD) into specific analog signal levels(i_(p), −i_(p) and 0). The reason for the conversion is that, since asignal amplitude in a digital circuit has a large allowance width, aconversion to a specific analog signal level is necessary. (For example,see: Floyd M. Gardner; “Phaselock Techniques”, 2nd edition, John Wiley &Sons, 1979; and Heinrich Meyr and Gerd Ascheid; Synchronization inDigital Communications”, vol. 1, John Wiley & Sons, 1990.)

As shown in FIG. 8A, a charge pump circuit comprises two currentsources. In this case, in order to simplify the model circuit, it isassumed that each of the current sources has the same current valueI_(p) Further, in order to simply describe an output current i_(p) ofthe charge pump circuit, a negative pulse width is introduced as shownin FIG. 8B. The logical signals UP and DOWN open/close switches S₁ andS₂, respectively. That is, the logical signal UP closes the switch S₁during a time period of positive pulse width τ and the logical signalDOWN closes the switch S₂ during a time period of negative pulse widthτ. Therefore, the output current i_(p) is represented, during the timeperiod of pulse width τ, by the following equation.

i _(p) =I _(p) sgn(τ)  (2.1.1)

Otherwise, the output current i_(p) is as follows.

i _(p)=0  (2.1.2)

(For example, see: Mark Van Paemel; “Analysis of Charge-Pump PLL: A NewModel”, IEEE Trans. Commun., vol. 42, pp. 2490-2498, 1994.)

In this case, sgn(τ) is a sign function. The function sgn(τ) takes avalue of +1 when τ is positive, and takes a value of −1 when τ isnegative. When the two switches S₁ and S₂ are open, no current flows.Therefore, the output node is in high impedance.

A loop filter converts a current i_(p) of the charge pump circuit intoan analog voltage value V_(CTRL). As shown in FIG. 9A, a first orderloop filter can be constructed when a resister R₂ and a capacitor C areconnected in series. When a constant current i_(p) given by theequations (2.1.1) and (2.1.2) is inputted to the filter, an electriccharge proportional to a time length is charged in the capacitor C. Thatis, as shown in FIG. 9B, the control voltage V_(CTRL) linearly changesduring the time period τ. In the other time period, the control voltageV_(CTRL) remains constant (for example, see the literature of Mark VanPaemel). $\begin{matrix}{{{V_{CTRL}(t)} = {{\frac{1}{C}{\int_{t_{0}}^{t}{{i_{P}(\tau)}\quad {\tau}}}} + {V_{CTRL}\left( t_{0} \right)}}},{{V_{CTRL}(t)} = {{I_{P}R_{2}} + {\frac{I_{P}}{C}\left( {t - t_{0}} \right)} + {V_{CTRL}\left( t_{0} \right)}}}} & (2.2)\end{matrix}$

The resistance value and the capacitance value of the loop filter areselected such that an attenuation coefficient and a natural frequencyare optimized. (For example, see: Jose Alvarez, Hector Sanchez,Gianfranco Gerosa and Roger Countryman; “A Wide-bandwidth Low-voltagePLL for Power PC Microprocessors”, IEEE J. Solid-State Circuits, vol.30, pp. 383-391, 1995; and Behzad Razavi; “Monolithic Phase-Locked Loopsand Clock Recovery Circuits: Theory and Design”, IEEE Press, 1996.) Inthe present invention, the loop filter is configured as a passive lagfilter as shown in FIG. 10 in accordance with a thesis by Ronald E. Bestlisted below. (See: Ronald E. Best; “Phase-Locked Loops”, 3rd edition,McGraw-Hill, 1997.) Because, as disclosed in this Ronald E. Best'spublication, the combination of a phase-frequency detector and a passivelag filter has infinite pull-in range and hold range, and hence there isno merit in using an other type of filter. In FIG. 10, C=250 pF, R₁=920Ω, and R₂=360 Ω are set. The VCO is constituted, as shown in FIG. 11, bythirteen stages of CMOS inverters IN-1, IN-2, . . . , IN-13. The powersupply voltage is 5 V.

The linear characteristic of the voltage controlled oscillator VCO isgiven by the following equation.

f _(VCO) =K _(VCO) V _(CTRL)  (2.3)

In this case, K_(VCO) is a gain of the VCO, and its units is Hz/V.

When the PLL is in synchronous state (a state that a rise edge of areference clock accords with a rise edge of a PLL clock), thephase-frequency detector outputs no signal. The charge pump circuit, theloop filter and the VCO provided in the rear stages of the PLL do notsend/receive signals and keep maintain the internal state unchanged. Onthe contrary, when a rise edge of a reference clock does not accord witha rise edge of a PLL clock (in asynchronous state), the phase-frequencydetector outputs an UP signal or a DOWN signal to change the oscillationfrequency of the VCO. As a result, the charge pump circuit, the loopfilter and the VCO provided in the rear stages of the PLL send/receivesignals and change into a corresponding state. Therefore, it could beunderstood, in order to measure an internal noise of the PLL circuit,that PLL circuit must be placed in a synchronous state. On the otherhand, in order to test a short-circuit failure or a delay failure of thePLL circuit, the PLL circuit must be moved into another state.

Now, a random jitter will be described.

A jitter on a clock appears as a fluctuation of a rise time and a falltime of a clock pulse series. For this reason, in the transmission of aclock signal, the receiving time or the pulse width of the clock pulsebecomes uncertain. (For example, see: Ron K. Poon; “Computer CircuitsElectrical Design”, Prentice-Hall, Inc, 1995.) FIG. 12 shows jitters ofa rise time period and a fall time period of a clock pulse series.

Any component in the blocks shown in FIG. 4 has a potential to cause ajitter. Among those components, the largest factors of a jitter are athermal noise and a shot noise of the inverters composing the VCO. (Forexample, see: Todd C. Weigandt, Beomsup Kim and Paul R. Gray; “Analysisof Timing Jitter in CMOS Ring Oscillators”, International Symposium onCircuits and System, 1994.) Therefore, the jitter generated from the VCOis a random fluctuation and does not depend on the input. In the presentinvention, assuming that the major jitter source is the VCO, it isconsidered that the measurement of a random jitter of an oscillationwaveform of the VCO is the most important problem to be solved.

In order to measure only a random jitter of an oscillation waveform ofthe VCO, it is necessary that the PLL circuit maintains the componentsother than the VCO to be inactive. Therefore, as mentioned above, it isimportant that a reference input signal to be supplied to the PLLcircuit strictly maintains a constant period so that the PLL circuitunder test does not induce a phase error. A concept of this measuringmethod is shown in FIG. 13.

As a preparation for discussing a phase noise, a zero crossing isdefined. Assuming that the minimum value −A of a cosine wave Acos(2πf₀t)is 0% and the maximum value +A thereof is 100%, a level of 50%corresponds to a zero amplitude. A point where the waveform crosses azero level is called a zero crossing.

A phase noise will be discussed with reference to, as an example, acosine wave generated from an oscillator. An output signal X_(IDEAL)(t)of an ideal oscillator is an ideal cosine wave having no distortion.

X _(IDEAL)(t)=A _(c) cos(2πf _(c) t+θ _(c))  (2.4)

In this case, A_(C) and f_(C) are nominal values of an amplitude and afrequency, respectively, and θ_(C) is an initial phase angle. When theoutput signal X_(IDEAL)(t) is observed in frequency domain, the outputsignal is measured as a line spectrum as shown in FIG. 14. In the actualoscillator, there are some differences from the nominal values. In thiscase, the output signal is expressed as follows.

X _(OSC)(t)=[A _(C)+ε(t)]cos(2πf _(C) t+θ _(C)+Δφ(t))  (2.5.1)

X _(OSC)(t)=A _(C) cos(2πf _(C) t+θ _(C)+Δφ(t))  (2.5.2)

In the above equations, ε(t) represents a fluctuation of an amplitude.In the present invention, the discussion will be made assuming that, asshown in the equation (2.5.2), the amplitude fluctuation ε(t) of theoscillator is zero. In the above equations, Δφ(t) represents a phasefluctuation. That is, Δφ(t) is a term for modulating the ideal cosinewave. The initial phase angle θ_(C) follows a uniform distribution inthe range of an interval (0,2π). On the other hand, the phasefluctuation Δφ(t) is a random data and follows, for example, a Gaussiandistribution. This Δφ(t) is called a phase noise.

In FIG. 15, an output signal X_(IDEAL)(t) of an ideal oscillator and anoutput signal X_(OSC)(t) of an actual oscillator are plotted. Comparingthose signals with one another, it can be seen that the zero crossing ofX_(OSC)(t) is changed due to Δφ(t).

On the other hand, as shown in FIG. 16, when the oscillation signalX_(OSC)(t) is transformed into frequency domain, the influence of aphase noise is observed as a spectrum diffusion in the proximity of thenominal frequency f_(c). Comparing FIG. 15 with FIG. 16, it can be saidthat frequency domain is easier to observe the influence of a phasenoise. However, even if the clock pulse shown in FIG. 12 is transformedinto frequency domain, the maximum value of the pulse width fluctuationcannot be estimated. Because, the transformation is a process foraveraging certain frequencies, and in the summing step of the process,the maximum value and the minimum value are mutually canceled.Therefore, in a peak-to-peak jitter estimating method which is an objectof the present invention, a process in time domain must be a nucleus ofthe method.

Here, it will be made clear that an additive noise at the referenceinput end to the PLL circuit is equal to an additive noise at the inputend of the loop filter. (See: Floyd M. Gardner; “Phaselock Techniques”,2nd edition, John Wiley & Sons, 1979; and John G. Proakis; “DigitalCommunications”, 2nd edition, McGraw-Hill, 1989.) FIG. 19 shows anadditive noise at the reference input end to the PLL circuit. In orderto simplify the calculation, it is assumed that a phase detector of thePLL circuit is a sine wave phase detector (mixer).

The PLL circuit is phase-synchronized with a given reference signalexpressed by the following equation (2.6).

X _(ref) =A _(C) cos(2πf _(C) t)  (2.6)

In this case, it is assumed that the following additive noise expressedby the equation (2.7) is added to this reference signal X^(ref).

X _(noise)(t)=n _(i)(t)cos(2πf _(C) t)−n _(q)(t)sin(2πf _(C) t)  (2.7)

X _(VCO)(t)=cos(2πf _(C) t+Δφ)  (2.8)

An oscillation waveform of the VCO expressed by the above equation (2.8)and the reference signal X_(ref)(t)+X_(noise)(t) are inputted to thephase detector to be converted to a difference frequency component.$\begin{matrix}{{x_{PD}(t)} = {{K_{PD}\left( {{\frac{A_{C}}{2}{\cos \left( {\Delta \quad \varphi} \right)}} + {\frac{n_{i}(t)}{2}{\cos \left( {\Delta \quad \varphi} \right)}} - \quad {\frac{n_{q}(t)}{2}{\sin \left( {\Delta \quad \varphi} \right)}}} \right)} = {\frac{K_{PD}}{2}{A_{C}\left\lbrack {{\cos \left( {\Delta \quad \varphi} \right)} + \left( {{\frac{n_{i}(t)}{A_{C}}{\cos \left( {\Delta \quad \varphi} \right)}} - \quad {\frac{n_{q}(t)}{A_{C}}{\sin \left( {\Delta \quad \varphi} \right)}}} \right)} \right\rbrack}}}} & (2.9)\end{matrix}$

In this case, K_(PD) is a gain of a phase comparator. Therefore, it canbe understood that the additive noise of the reference signal is equalto that an additive noise expressed by the following equation (2.10) isapplied to an input end of the loop filter. $\begin{matrix}{{x_{{noise},{LPF}}(t)} = {{\frac{n_{i}(t)}{A_{C}}{\cos \left( {\Delta \quad \varphi} \right)}} - \quad {\frac{n_{q}(t)}{A_{C}}{\sin \left( {\Delta \quad \varphi} \right)}}}} & (2.10)\end{matrix}$

FIG. 18 shows an additive noise at the input end of the loop filter. Ifa power spectrum density of the additive noise at the reference inputend of the PLL circuit is assumed to be N₀[V²/Hz], the power spectrumdensity G_(nn)(f) of the additive noise at the input end of this loopfilter is, from the equation (2.10), expressed by the following equation(2.11). $\begin{matrix}{{G_{nn}(f)} = {\frac{2N_{0}}{A_{C}^{2}}\left\lbrack {V^{2}/{Hz}} \right\rbrack}} & (2.11)\end{matrix}$

Moreover, it can be seen from the equation (2.9) that when a phasedifference Ad between the oscillation waveform of the VCO and thereference signal becomes π/2, an output of the phase detector becomeszero. That is, if a sine wave phase detector is used, when the phase ofthe VCO is shifted by 90 degrees from the phase of the reference signal,the VCO is phase-synchronized with the reference signal. Further, inthis calculation, the additive noise is neglected.

Next, using a model of equivalent additive noise shown in FIG. 17, anamount of jitter produced by an additive noise will be made clear. (See:Heinrich Meyr and Gerd Ascheid; “Synchronization in DigitalCommunications”, vol. 1, John Wiley & Sons, 1990.) In order to simplifythe expression, assuming θ_(i)=0, the phase θ₀ of the output signalcorresponds to an error. An phase spectrum of the oscillation waveformof the VCO is expressed by the following equation (2.12).$\begin{matrix}{{G_{\theta_{0}\theta_{0}}(f)} = {{{H(f)}}^{2}{G_{nn}(f)}}} & (2.12)\end{matrix}$

In this case, H(f) is a transfer function of the PLL circuit.$\begin{matrix}{{H(s)} = {\frac{\theta_{0}(s)}{\theta_{i}(s)} = \frac{K_{VCO}K_{PD}{F(s)}}{s + {K_{VCO}K_{PD}{F(s)}}}}} & (2.13)\end{matrix}$

Since a phase error is −θ₀, a variance of the phase error is given bythe following equation (2.14). $\begin{matrix}{\sigma_{\Delta \quad \varphi}^{2} = {\frac{1}{\pi}{\int_{0}^{\infty}{{{H(f)}}^{2}{G_{nn}(f)}\quad {f}}}}} & (2.14)\end{matrix}$

Substituting the equation (2.11) for the equation (2.14), the followingtwo equations are obtained. $\begin{matrix}{\sigma_{\Delta \quad \varphi}^{2} = {\frac{2N_{0}}{A_{c}^{2}}B_{e}}} & \text{(2.15.1)} \\{\sigma_{\Delta \quad \varphi}^{2} = \frac{1}{\frac{\left( \frac{A_{c}}{\sqrt{2}} \right)^{2}}{N_{0}B_{e}}}} & \text{(2.15.2)}\end{matrix}$

That is, if a signal to noise ratio of the loop is large, a phase noisebecomes small. In this case, B_(e) is an equivalent noise band width ofthe loop.

As described above, an additive noise${\left( \frac{A_{c}}{\sqrt{2}} \right)/N_{0}}B_{e}$

at the reference input end of the a PLL circuit or an additive noise atthe input end of the loop filter is observed as an output phase noise,which is a component passed through a lowpass filter corresponding tothe loop characteristic. The power of a phase noise is inverselyproportional to a signal to noise ratio of the PLL loop.

Next, a discussion will be made as to how a phase fluctuation due to aninternal noise of the VCO influences a phase of output signal of thePLL. (See: Heinrich Meyr and Gerd Ascheid; “Synchronization in DigitalCommunications”, vol. 1, John Wiley & Sons, 1990.) An output signal ofthe VCO is assumed to be expressed by the following equation (2.16).

 X _(VCO, noise) =A _(C) cos(2πf _(c) t+θ _(P)(t)+ψ(t))  (2.16)

In this case, θ_(P)(t) is a phase of an ideal VCO. An internal thermalnoise or the like generates ψ(t). The generated ψ(t) is an internalphase noise and randomly fluctuates the phase of the VCO. FIG. 19 showsan internal phase noise model of the VCO. A phase θ_(P)(S) at the outputend of the ideal VCO is given by a equation (2.17). $\begin{matrix}{{\theta_{p}(s)} = {K_{PD}K_{VCO}\quad \frac{F(s)}{s}{\Phi (s)}}} & (2.17)\end{matrix}$

In this case, Φ(t) is a phase error and corresponds to an output of thephase detector.

Φ(S)=θ_(i)(S)−θ₀(S)=θ_(i)(S)−(θ_(P)(S)+ψ(S))  (2.18)

Substituting θ_(P)(S) of the equation (2.17) for that of the equation(2.18), the following equation (2.19) is obtained. $\begin{matrix}{{\Phi (s)} = {{\theta_{i}(s)} - \left\lbrack {{\frac{K_{PD}K_{VCO}{F(s)}}{s}{\Phi (s)}} + {\psi (s)}} \right\rbrack}} & (2.19)\end{matrix}$

The following equation (2.20.1) can be obtained by rearranging the aboveequation (2.19). $\begin{matrix}{{\Phi (s)} = {\frac{1}{1 + \frac{K_{PD}K_{VCO}{F(s)}}{s}}\left( {{\theta_{i}(s)} - {\psi (s)}} \right)}} & \text{(2.20.1)}\end{matrix}$

Substituting the equation (2.13) for the equation (2.20.1), thefollowing equation (2.20.2) is obtained.

Φ(S)=(1−H(S))(θ_(i)(S)−ψ(S))  (2.20.2)

Therefore, a phase fluctuation due to an internal noise of the VCO isexpressed by the following equation (2.21). $\begin{matrix}{\sigma_{\Phi}^{2} = {\frac{1}{\pi}{\int_{0}^{\infty}{{{1 - {H(f)}}}^{2}{G_{\psi \quad \psi}(f)}{f}}}}} & \text{(2.21)}\end{matrix}$

That is, an internal phase noise of the VCO is observed as a phase noiseof an output signal of the PLL circuit, which is a component passedthrough a highpass filter. This highpass filter corresponds to a phaseerror transfer function of the loop.

As stated above, an internal thermal noise of the VCO becomes a phasenoise of an oscillation waveform of the VCO. Further, a component passedthrough the highpass filter corresponding to a loop phase error isobserved as an output phase noise.

An additive noise of the PLL circuit and/or an internal thermal noise ofthe VCO is converted to a phase noise of an oscillation waveform of theVCO. An additive noise of the PLL circuit and/or an internal thermalnoise of the VCO is observed, correspondingly to the path from a blockgenerating a noise through the output of the PLL circuit, as a phasenoise of a low frequency component or a high frequency component.Therefore, it can be seen that a noise of the PLL circuit has an effectto give a fluctuation to a phase of an oscillation waveform of the VCO.This is equivalent to a voltage change at the input end of the VCO. Inthe present invention, an additive noise is applied to the input end ofthe VCO to randomly modulate the phase of a waveform of the VCO so thata jitter is simulated. FIG. 20 shows a method of simulating a jitter.

Next, a method of measuring a jitter of a clock will be explained. Apeak-to-peak jitter is measured in time domain and an RMS jitter ismeasured in frequency domain. Each of those conventional jittermeasuring methods requires approximately 10 minutes of test time. On theother hand, in the case of a VLSI test, only approximately 100 msec oftest time is allocated to one test item. Therefore, the conventionalmethod of measuring a jitter cannot be applied to a test in the VLSIproduction line.

In the study of the method of measuring a jitter, the zero crossing isan important concept. From the view point of period measurement, arelationship between the zero crossings of a waveform and the zerocrossings of the fundamental waveform of its fundamental frequency willbe discussed. It will be proven that “the waveform of its fundamentalfrequency contains the zero-crossing information of the originalwaveform”. In the present invention, this characteristic of thefundamental waveform is referred to as “theorem of zero crossing”. Anexplanation will be given on an ideal clock waveform X_(d50%)(t) shownFIG. 21, as an example, having 50% duty cycle. Assuming that a period ofthis clock waveform is T₀, the Fourier transform of the clock waveformis given by the following equation (3.1). (For example, refer to areference literature c1.) $\begin{matrix}{{S_{{d50}\%}(f)} = {\sum\limits_{k = {- \infty}}^{+ \infty}{\frac{2\quad {\sin \left( \frac{\pi \quad k}{2} \right)}}{k}{\delta \left( {f - {kf}_{0}} \right)}}}} & \text{(3.1)}\end{matrix}$

That is, a period of the fundamental is equal to a period of the clock.

T ₀=1/fδ(f−f ₀)  (3.2)

When the fundamental waveform is extracted, its zero crossingscorresponds to the zero crossings of the original clock waveform.Therefore, a period of a clock waveform can be estimated from the zerocrossings of its fundamental waveform. Therefore, the estimationaccuracy is not improved even if some harmonics are added to thefundamental waveform. However, harmonics and an estimation accuracy of aperiod will be verified later.

Next, Hilbert transform and an analytic signal will be explained (forexample, refer to a reference literature c2).

As can be seen from the equation (3.1), when the Fourier transform ofthe waveform X_(a)(t) is calculated, a power spectrum S_(aa)(f) rangingfrom negative frequencies through positive frequencies can be obtained.This is called a two-sided power spectrum. The negative frequencyspectrum is a mirror image of the positive frequency spectrum about anaxis of f=0. Therefore, the two-sided power spectrum is symmetry aboutthe axis of f=0, i.e., S_(aa)(−f)=S_(aa)(f). However, the spectrum ofnegative frequencies cannot be observed. There can be defined a spectrumG_(aa)(f) in which negative frequencies are cut to zero and, instead,observable positive frequencies are doubled. This is called one-sidedpower spectrum.

G _(aa)(f)=2S _(aa)(f) f>0G _(aa)(f)=0f<0  (3.3.1)

G _(aa)(f)=S _(aa)(f)[1+sgn(f)]  (3.3.2)

In this case, sgn(f) is a sign function, which takes a value of +1 Whenf is positive and takes a value of −1 when f is negative. This one-sidedspectrum corresponds to a spectrum of an analytic signal z(t). Theanalytic signal z(t) can be expressed in time domain as follows.

z(t)=x _(a) (t)+j{circumflex over (x)} _(a)(t)  (3.4)

$\begin{matrix}{{{\hat{x}}_{a}(t)} = {{H\left\lbrack {x_{a}(t)} \right\rbrack} = {\frac{1}{\pi}{\int_{- \infty}^{+ \infty}{\frac{x_{a}(\tau)}{t - \tau}{\tau}}}}}} & \text{(3.5)}\end{matrix}$

The real part corresponds to the original waveform X_(a)(t). Theimaginary part {circumflex over (X)}_(a)(t) is given by the Hilberttransform of the original waveform. As shown by the equation (3.5), theHilbert-transformed {circumflex over (X)}_(a)(t) of a waveform X_(a)(t)is given by a convolution of the X_(a)(t) and 1/πt.

Let's obtain the Hilbert transform of a waveform handled in the presentinvention. First, the Hilbert transform of a cosine wave is derived.${H\left\lbrack {\cos \left( {2\quad \pi \quad f_{0}t} \right)} \right\rbrack} = {{{- \frac{1}{\pi}}{\int_{- \infty}^{+ \infty}{\frac{\cos \left( {2\quad \pi \quad f_{0}\tau} \right)}{\tau - 1}{\tau}}}} = {{{- \frac{1}{\pi}}{\int_{- \infty}^{+ \infty}{\frac{\cos \left( {2\quad \pi \quad {f_{0}\left( {y + t} \right)}} \right)}{y}{y}}}} = {- {\frac{1}{\pi}\left\lbrack {{{\cos \left( {2\quad \pi \quad f_{0}t} \right)}{\int_{- \infty}^{+ \infty}{\frac{\cos \left( {2\quad \pi \quad f_{0}y} \right)}{y}{y}}}} - {{\sin \left( {2\quad \pi \quad f_{0}t} \right)}{\int_{- \infty}^{+ \infty}{\frac{\sin \left( {2\quad \pi \quad f_{0}y} \right)}{y}{y}}}}} \right\rbrack}}}}$

Since the integral of the first term is equal to zero and the integralof the second term is π, the following equation (3.6) is obtained.

H[cos(2πf ₀ t)]=sin(2πf ₀ t)  (3.6)

Similarly, the following equation (3.7) is obtained.

H[sin(2πf ₀ t)]=−cos(2πf ₀ t)  (3.7)

Next, the Hilbert transform of a square wave corresponding to a clockwaveform will be derived (for example, refer to a reference literaturec3). The Fourier series of an ideal clock waveform shown in FIG. 21 isgiven by the following equation (3.8). $\begin{matrix}{{x_{{d50}\%}(t)} = {\frac{1}{2} + {\frac{2}{\pi}\left\lbrack {{\cos \frac{2\quad \pi}{T_{0}}t} - {\frac{1}{3}\cos \quad 3\frac{2\quad \pi}{T_{0}}t} + {\frac{1}{5}\cos \quad 5\frac{2\quad \pi}{T_{0}}t} - \ldots}\quad \right\rbrack}}} & \text{(3.8)}\end{matrix}$

The Hilbert transform is given, using the equation (3.6), by thefollowing equation (3.9). $\begin{matrix}{{H\left\lbrack {x_{{d50}\%}(t)} \right\rbrack} = {\frac{2}{\pi}\left\lbrack {{\sin \frac{2\quad \pi}{T_{0}}t} - {\frac{1}{3}\sin \quad 3\frac{2\quad \pi}{T_{0}}t} + {\frac{1}{5}\quad \sin \quad 5\frac{2\quad \pi}{T_{0}}t} - \ldots}\quad \right\rbrack}} & \text{(3.9)}\end{matrix}$

FIG. 22 shows examples of a clock waveform and its Hilbert transform.Those waveforms are based on the summation up to the 11^(th)-orderharmonics, respectively. The period T₀ in this example is 20 nsec.

An analytic signal z(t) is introduced by J. Dugundji to uniquely obtainan envelope of a waveform. (For example, refer to a reference literaturec4.) If an analytic signal is expressed in a polar coordinate system,the following equations (3.10.1), (3.10.2) and (3.10.3) are obtained.

z(t)=A(t)e ^(jΘ(t))  (3.10.1)

A(t)={square root over (x _(a) ² (t)+{circumflex over (x)})} _(a)²(t)  (3.10.2)

$\begin{matrix}{{\Theta (t)} = {\tan^{- 1}\left\lbrack \frac{{\hat{x}}_{a}(t)}{x_{a}(t)} \right\rbrack}} & \text{(3.10.3)}\end{matrix}$

In this case, A(t) represents an envelope of X_(a)(t). For this reason,z(t) is called pre-envelope by J. Dugundji. Further, Θ(t) represents aninstantaneous phase of X_(a)(t). In the method of measuring a jitteraccording to the present invention, a method of estimating thisinstantaneous phase is the nucleus.

If a measured waveform is handled as a complex number, its envelope andinstantaneous phase can simply be obtained. Hilbert transform is a toolfor transforming a waveform to an analytic signal. An analytic signalcan be obtained by the procedure of the following Algorithm 1.

Algorithm 1 (Procedure for transforming a real waveform to an analyticsignal):

1. A waveform is transformed to a frequency domain using fast Fouriertransform;

2. Negative frequency components are cut to zero and positive frequencycomponents are doubled; and

3. The spectrum is transformed to a time domain using inverse fastFourier transform.

Next, a phase unwrap method for converting a phase to a continuous phasewill briefly be described.

The result of Fourier transform of a time waveform X_(a)(n) is assumedto be S_(a)(e^(jω)). The phase unwrap method is a method proposed toobtain a complex cepstrum. (For example, refer to a reference literaturec5.) When a complex logarithmic function log(z) is defined as anarbitrary complex number satisfying e^(log(z))=z, the following equation(3.11) can be obtained. (For example, refer to a reference literaturec6.)

log (z)=log|z|+jARG(z)  (3.11)

The Fourier transform of the time waveform X_(a)(n)is assumed to beS_(a)(e^(jω)). When its logarithmic magnitude spectrum log|S_(a)(e^(jω))| and phase spectrum ARG[S_(a)(e^(jω))] correspond to areal part and an imaginary part of a complex spectrum, respectively, andinverse Fourier transform is applied, a complex cepstrum C_(a)(n) can beobtained. $\begin{matrix}\begin{matrix}{{c_{a}(n)} = {\frac{1}{2\quad \pi}{\int_{- \pi}^{+ \pi}{{\log \left\lbrack {S_{a}\left( ^{j\quad \omega} \right)} \right\rbrack}^{j\quad \omega \quad n}{\omega}}}}} \\{= {\frac{1}{2\quad \pi}{\int_{- \pi}^{+ \pi}{\left\{ {{\log {{S_{a}\left( ^{j\quad \omega} \right)}}} + {j\quad {{ARG}\left\lbrack {S_{a}\left( ^{j\quad \omega} \right)} \right\rbrack}}} \right\} ^{j\quad \omega \quad n}{\omega}}}}}\end{matrix} & \text{(3.12)}\end{matrix}$

In this case, ARG represents the principal value of the phase. Theprincipal value of the phase lies in the range [−π,+π]. There existdiscontinuity points at −π and +π in the phase spectrum of the 2nd term.Since an influence of those discontinuity points diffuses throughoutentire time domain by the application of inverse Fourier transform, acomplex cepstrum cannot accurately be estimated. In order to convert aphase to a continuous phase, an unwrapped phase is introduced. Anunwrapped phase can be uniquely given by integrating a derived functionof a phase.

$\begin{matrix}{{\arg\left\lbrack {S_{a}\left( ^{j\quad \omega} \right)} \right\rbrack} = {\int_{0}^{\omega}{\frac{{{ARG}\left\lbrack {S_{a}\left( ^{j\quad \eta} \right)} \right\rbrack}}{\eta}{\eta}}}} & \text{(3.13.1)} \\{{\arg\left\lbrack {S_{a}\left( ^{j\quad 0} \right)} \right\rbrack} = 0} & \text{(3.13.2)}\end{matrix}$

Where, arg represents an unwrapped phase. An algorithm for obtaining anunwrapped phase by removing discontinuity points from a phase spectrumin frequency domain has been developed by Ronald W. Schafer and DonaldG. Childers (for example, refer to a reference literature c7).

Algorithm 2:

ARG(0)=0, C(0)=0  1

$\begin{matrix}{{C(k)} = \left\{ \begin{matrix}{{{C\left( {k - 1} \right)} - {2\quad \pi}},} & {{{{if}\quad {{ARG}(k)}} - {{ARG}\left( {k - 1} \right)}} > \pi} \\{{{C\left( {k - 1} \right)} + {2\quad \pi}},} & {{{{if}\quad {{ARG}(k)}} - {{ARG}\left( {k - 1} \right)}} < \pi} \\{{C\left( {k - 1} \right)},} & {{otherwise}.}\end{matrix} \right.} & 2\end{matrix}$

 arg(k)=ARG(k)+C(k)  3

An unwrapped phase will be obtained from the above Algorithm 2. First, ajudgement is made, by obtaining differences between main values ofadjacent phases, to see if there is a discontinuity point. If there is adiscontinuity point, ±2π, is added to the main value to remove thediscontinuity point from the phase spectrum (refer to the referenceliterature c7).

In the above algorithm 2, it is assumed that a difference betweenadjacent phases is smaller than π. That is, a resolution for observing aphase spectrum is required to be sufficiently small. However, at afrequency in the proximity of a pole (a resonance frequency), the phasedifference between the adjacent phases is larger than π. If a frequencyresolution for observing a phase spectrum is rough, it cannot bedetermined whether or not a phase is increased or decreased by equal toor more than 2π. As a result, an unwrapped phase cannot be accuratelyobtained. This problem has been solved by Jose M. Tribolet. That is,Jose M. Tribolet proposed a method wherein the integration of thederived function of a phase in the equation (3.12) is approximated by anumerical integration based on a trapezoidal rule and a division widthof the integrating section is adaptively subdivided to fine pieces untilan estimated phase value for determining whether or not a phase isincreased or decreased by equal to or more than 2π is obtained (forexample, refer to a reference literature c8). In such a way, an integerl of the following equation (3.14) is found.

arg[S _(a)(e ^(jΩ))]=ARG[S _(a)(e ^(jΩ))]+2πl(Ω)  (3.14)

The Tribolet's algorithm has been expanded by Kuno P. Zimmermann to aphase unwrap algorithm in time domain (for example, refer to a referenceliterature c9).

In the present invention, the phase unwrap is used to convert aninstantaneous phase waveform in time domain into a continuous phaseexcept discontinuity points at −π and +π in the instantaneous phasewaveform. A sampling condition for uniquely performing the phase unwrapin time domain will be discussed later.

Next, a linear trend estimating method to be utilized to obtain a linearphase from a continuous phase will briefly be described (for example,refer to reference literatures c10 and c11).

The target of the linear trend estimating method is to find a linearphase g(x) adaptable to a phase data y_(i).

g(x)=a+bx  (3.15)

In this case, “a” and “b” are the constants to be found. A square errorR between g(x_(i)) and each data (x_(i), y_(i)) is given by thefollowing equation (3.16). $\begin{matrix}{R = {\sum\limits_{i = 1}^{L}\left( {y_{i} - a - {bx}_{i}} \right)^{2}}} & \text{(3.16)}\end{matrix}$

In this case, L is the number of phase data. A linear phase forminimizing the square error is found. A partial differentiation of theequation (3.16) with respect to each of the unknown constants a and b iscalculated and the result is put into zero. Then the following equations(3.17.1) and (3.17.2) can be obtained. $\begin{matrix}{\frac{\partial R}{\partial a} = {{\sum\limits_{i = 1}^{L}\left( {y_{i} - a - {bx}_{i}} \right)} = 0}} & \text{(3.17.1)} \\{\frac{\partial R}{\partial b} = {{\sum\limits_{i = 1}^{L}{x_{i}\left( {y_{i} - a - {bx}_{i}} \right)}} = 0}} & \text{(3.17.2)}\end{matrix}$

Those equations are transformed to obtain the following equation (3.18).$\begin{matrix}{{\begin{bmatrix}L & {\Sigma x}_{i} \\{\Sigma x}_{i} & {\Sigma x}_{i}^{2}\end{bmatrix}\begin{bmatrix}a \\b\end{bmatrix}} = \begin{bmatrix}{\Sigma \quad y_{i}} \\{\Sigma \quad x_{i}y_{i}}\end{bmatrix}} & \text{(3.18)}\end{matrix}$

Therefore, the following equation (3.19) can be obtained.$\begin{matrix}{\begin{bmatrix}a \\b\end{bmatrix} = {{\frac{1}{{L\quad \Sigma \quad x_{i}^{2}} - \left( {\Sigma \quad x_{i}} \right)^{2}}\begin{bmatrix}{\Sigma \quad x_{i}^{2}} & {{- \Sigma}\quad x_{i}} \\{{- \Sigma}\quad x_{i}} & L\end{bmatrix}}\begin{bmatrix}{\Sigma \quad y_{i}} \\{\Sigma \quad x_{i}y_{i}}\end{bmatrix}}} & \text{(3.19)}\end{matrix}$

That is, a linear phase can be estimated from the following equations(3.20.1) and (3.20.2). $\begin{matrix}{a = \frac{{\Sigma \quad x_{i}^{2}\Sigma \quad y_{i}} - {\Sigma \quad x_{i}\Sigma \quad x_{i}y_{i}}}{{L\quad \Sigma \quad x_{i}^{2}} - \left( {\Sigma \quad x_{i}} \right)^{2}}} & \text{(3.20.1)} \\{b = \frac{{L\quad \Sigma \quad x_{i}y_{i}} - {\Sigma \quad x_{i}\Sigma \quad y_{i}}}{{L\quad \Sigma \quad x_{i}^{2}} - \left( {\Sigma \quad x_{i}} \right)^{2}}} & \text{(3.20.2)}\end{matrix}$

In the present invention, when a linear phase is estimated from acontinuous phase, a linear trend estimating method is used.

As apparent from the above discussion, in the conventional method ofmeasuring a jitter, a peak-to-peak jitter is measured in time domainusing an oscilloscope and an RMS jitter is measured in frequency domainusing a spectrum analyzer.

In the method of measuring a jitter in time domain, a peak-to-peakjitter J_(PP) of a clock signal is measured in time domain. FIGS. 23 and24 show a measured example of a peak-to-peak jitter measured using anoscilloscope and the measuring system, respectively. A clock signalunder test is applied to a reference input of the phase detector. Inthis case, the phase detector and the signal generator compose aphase-locked loop. A signal of the signal generator is synchronized withthe clock signal under test and is supplied to an oscilloscope as atrigger signal. In this example, a jitter of rise edge of the clocksignal is observed. A square zone is used to specify a level to becrossed by the signal. A jitter is measured as a varying component oftime difference between “a time point when the clock signal under testcrosses the specified level” and “a reference time point given by thetrigger signal”. This method requires a longer time period for themeasurement. For this reason, the trigger signal must bephase-synchronized with the clock signal under test so that themeasurement is not influenced by a frequency drift of the clock signalunder test.

A measurement of a jitter in time domain corresponds to a measurement ofa fluctuation of a time point when a level is crossed by the signal.This is called, in the present invention, a zero crossing method. Sincea change rate of a waveform is maximum at the zero crossing, a timingerror of a time point measurement is minimum at the zero crossing.$\begin{matrix}{{\Delta \quad t} = {{\frac{\Delta \quad A}{{A2}\quad \pi \quad f_{0}{\sin \left( {2\quad \pi \quad f_{0}t} \right)}}} \geq \frac{\Delta \quad A}{2\quad \pi \quad f_{0}A}}} & \text{(3.21)}\end{matrix}$

In FIG. 25(a), the zero crossing is indicated by each of small circles.A time interval between a time point ti that a rise edge crosses a zeroamplitude level and a time point t_(i+2) that a next rise edge crosses azero amplitude level gives a period of this cosine wave. FIG. 25(b)shows an instantaneous period P_(inst) obtained from the zero crossing(found from adjacent zero crossings t_(i+1) and t_(i+2)). Ainstantaneous frequency f_(inst) is given by an inverse number ofP_(inst).

P _(inst)(t _(i+2))=t _(i+2) −t _(i) , P _(inst)(t _(i+2))=2(t _(i+2) −t_(i+1))  (3.22.1)

$\begin{matrix}{{f_{inst}\left( t_{i + 2} \right)} = \frac{1}{p_{inst}\left( t_{i + 2} \right)}} & \text{(3.22.2)}\end{matrix}$

Problems in measuring a jitter in time domain will be discussed. Inorder to measure a jitter, a rise edge of a clock signal under testX_(C)(t) is captured, using an oscilloscope, at a timing of the zerocrossing.

x _(c)(t)=A _(c) cos(2πf _(c) t+θ _(c)+Δφ(t))  (3.23)

This means that only X_(C)(t) satisfying the next condition of phaseangle given by the following equation (3.24) can be collected.$\begin{matrix}{{{2\quad \pi \quad f_{0}t_{\frac{3\quad \pi}{2}}} + \theta_{c} + {\Delta \quad {\varphi \left( t_{\frac{3\quad \pi}{2}} \right)}}} = {{{\pm 2}\quad m\quad \pi} + \frac{3\quad \pi}{2}}} & \text{(3.24)}\end{matrix}$

A probability density function of a sample corresponding to the zerocrossing of a rise edge is given by the following equation (3. 25). (Forexample, refer to a reference literature c10.) $\begin{matrix}{\frac{1}{2\quad \pi \sqrt{A_{c}^{2} - {x_{c}^{2}(t)}}}}_{{x_{c}{(t)}} = 0} & \text{(3.25)}\end{matrix}$

Therefore, a time duration required for randomly sampling a clock signalunder test to collect phase noises$\Delta \quad {\varphi \left( t_{\frac{3\quad \pi}{2}} \right)}$

of N points is given by the following equation (3.26).

(2πA _(c)) (NT ₀)  (3.26)

That is, since only zero crossing samples can be utilized for a jitterestimation, at least (2πA_(C)) times of test time period is requiredcompared with an usual measurement.

As shown in FIG. 26, the magnitude of a set of phase noises which can besampled by the zero crossing method is smaller than an entire set ofphase noises. Therefore, a peak-to-peak jitter J_(PP 3π/2) which can beestimated is equal to or smaller than a true peak-to-peak jitter J_(PP).$\begin{matrix}{J_{PP} = {{{\max\limits_{k}\left( {\Delta \quad {\varphi (k)}} \right)} - {\min\limits_{l}{\left( {{\Delta\varphi}(l)} \right)J_{{PP},{3{\pi/2}}}}}} \leq J_{PP}}} & (3.27)\end{matrix}$

The worst drawback of the zero crossing method is that a time resolutionof the period measurement cannot be selected independently on a periodof a signal under test. The time resolution of this method is determinedby a period of the signal under test, i.e., the zero crossing. FIG. 27is a diagram in which the zero crossings of the rise edges are plottedon a complex plane. The sample in the zero crossing method is only onepoint indicated by an arrow, and the number of samples per period cannotbe increased. When a number n_(i) is given to the zero crossing of arise edge, the zero crossing method measures a phase differenceexpressed by the following equation (3.28).

n _(i)(2π)  (3.28)

As a result, an instantaneous period measured by the zero crossingmethod comes to, as shown in FIG. 25(b), a rough approximation obtainedby use of a step function.

In 1988, David Chu invented a time interval analyzer (for example, referto reference literatures c12 and c13). In the time interval analyzer,when integer values n_(i) of the zero crossings n_(i)(2π) of the signalunder test are counted, the elapsed time periods t_(i) are alsosimultaneously counted. By this method, the time variation of the zerocrossing with respect to the elapsed time period could be plotted.Further, by using (t_(i), n_(i)), a point between measured data cansmoothly be interpolated by spline functions. As a result, it was madepossible to observe an instantaneous period approximated in higherorder. However, it should be noted that David Chu's time intervalanalyzer is also based on the zero crossing measurement of a signalunder test. Although the interpolation by spline functions makes iteasier to understand the physical meaning, the fact is that only thedegree of approximation of an instantaneous period is increased.Because, the data existing between the zero crossings have not beenstill measured. That is, the time interval analyzer cannot either exceedthe limit of the zero crossing method. An opposite example forinterpolating the instantaneous data will be discussed later.

Next, a method of measuring a jitter in frequency domain will bedescribed.

An RMS jitter J_(RMS) of a clock signal is measured in frequency domain.FIGS. 28 and 29 show an example of an RMS jitter measured by using aspectrum analyzer and a measuring system using a spectrum analyzer,respectively. A clock signal under test is inputted to a phase detectoras a reference frequency. In this case, the phase detector and thesignal generator compose a phase-locked loop. A phase difference signalbetween the clock signal under test detected by the phase detector andthe signal from the signal generator is inputted to the spectrumanalyzer to observe a phase noise spectrum density function. An areabelow the phase noise spectrum curve shown in FIG. 28 corresponds to anRMS jitter J_(RMS). The frequency axis expresses the offset frequenciesfrom the clock frequency. That is, zero (0) Hz corresponds to the clockfrequency.

A phase difference signal Δφ(t) between the clock signal under testX_(C)(t) expressed by the equation (3.23) and a reference signalexpressed by the following equation (3.29) is outputted from the phasedetector.

x _(ref)(t)=A cos(2πf _(c) t+θ ₀)  (3.29)

At this point in time, since the reference signal being applied to aphase-locked loop circuit (PLL circuit) under test has a constantperiod, the phase difference signal Δφ(t) corresponds to a phase noisewaveform. When the phase difference signal Δφ(t) is observed during afinite time period T and is transformed into time domain, a phase noisepower spectrum density function G_(ΔφΔφ)(f) can be obtained.$\begin{matrix}{{S_{\Delta \quad \varphi}(f)} = {\int_{0}^{T}{\Delta \quad {\varphi (t)}^{{- 2}\quad \pi \quad f\quad t}{t}}}} & \text{(3.30)} \\{{G_{\Delta \quad \varphi \quad \Delta \quad \varphi}(f)} = {\lim\limits_{T\rightarrow\infty}{\frac{2}{T}{E\left\lbrack {{S_{\Delta \quad \varphi}(f)}}^{2} \right\rbrack}}}} & \text{(3.31)}\end{matrix}$

From Parseval's theorem, a mean square value of a phase noise waveformis given by the following equation (3.32). (For example, refer to areference literature c14.) $\begin{matrix}{{{E\left\lbrack {\Delta \quad {\varphi^{2}(t)}} \right\rbrack} \equiv {\lim\limits_{T\rightarrow\infty}{\frac{1}{T}{\int_{0}^{T}{\Delta \quad {\varphi^{2}(t)}{t}}}}}} = {\int_{0}^{\infty}{{G_{\Delta \quad \varphi \quad \Delta \quad \varphi}(f)}{f}}}} & \text{(3.32)}\end{matrix}$

That is, it can be understood that by measuring a sum of the powerspectrum, a mean square value of a phase noise waveform can beestimated. A positive square root of the mean square value (an effectivevalue) is called RMS (a root mean square) jitter J_(RMS).$\begin{matrix}{J_{RMS} = \sqrt{\int_{0}^{f_{MAX}}{{G_{\Delta \quad \varphi \quad \Delta \quad \varphi}(f)}{f}}}} & \text{(3.33)}\end{matrix}$

When a mean value is zero, a mean square value is equivalent to avariance, and an RMS jitter is equal to a standard deviation.

As shown in FIG. 28, J_(RMS) can be accurately approximated to a sum ofG_(ΔφΔφ)(f) in the proximity of the clock frequency (for example, referto a reference literature c15). Actually, in the equation (3.33), theupper limit value f_(MAX) of the frequency of G_(ΔφΔφ)(f) to be summedis (2f_(C)−ε). Because, if G_(ΔφΔφ)(f) is summed in the frequency rangewider than the clock frequency, the harmonics of the clock frequency areincluded in J_(RMS).

In a measurement of an RMS jitter in frequency domain, there arerequired a phase detector, a signal generator whose phase noise is smalland a spectrum analyzer. As can be understood from the equation (3.33)and FIG. 28, a phase noise spectrum is measured by frequency-sweeping alow frequency range. For this reason, the measuring method requires ameasurement time period of approximately 10 minutes, and cannot beapplied to the test of a microprocessor. In addition, in the measurementof an RMS jitter in frequency domain, a peak-to-peak jitter cannot bemeasured since the phase information has been lost.

As described above, in the conventional method of measuring a jitter, apeak-to-peak jitter is measured in time domain using an oscilloscope.The basic method of measuring a jitter in time domain is the zerocrossing method. The biggest drawback of this method is that a timeresolution of a period measurement cannot be made fine independently onthe period of a signal under test. For this reason, a time intervalanalyzer for simultaneously counting the integer values n_(i) of thezero crossings of the signal under test n_(i)(2π) and the elapsed timeperiods t_(i) was invented. However, the data existing between the zerocrossings cannot be measured. That is, the time interval analyzer alsocannot exceed the limit of the zero crossing method.

On the other hand, an RMS jitter is measured in frequency domain using aspectrum analyzer. Since the phase information has been lost, apeak-to-peak jitter cannot be estimated.

In addition, either case of measuring a jitter in time domain ormeasuring an RMS jitter in frequency domain requires a measurement timeof approximately 10 minutes. In a test of a VLSI, a testing time of onlyapproximately 100 msec is allocated to one test item. Therefore, thereis a serious drawback in the conventional method of measuring a jitterthat the method cannot be applied to a test of a VLSI in themanufacturing process thereof.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an apparatus for anda method of measuring a jitter wherein a peak-to-peak jitter can bemeasured in a short test time of approximately 100 msec or so.

It is another object of the present invention to provide an apparatusfor and a method of measuring a jitter wherein data obtained from theconventional RMS jitter measurement or the conventional peak-to-peakjitter measurement can be utilized.

In order to achieve the above objects, in one aspect of the presentinvention, there is provided an apparatus for measuring a jittercomprising: a signal processing circuit for transforming a clockwaveform X_(C)(t) into an analytic signal using Hilbert transform andestimating a varying term Δφ(t) of an instantaneous phase of thisanalytic signal.

In another aspect of the present invention, there is provided a methodof measuring a jitter comprising the steps of: transforming a clockwaveform X_(C)(t) into an analytic signal using Hilbert transform; andestimating a varying term Δφ(t) of an instantaneous phase of thisanalytic signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing a relationship between a clock period of amicrocomputer and an RMS jitter;

FIG. 2 is a diagram showing a Pentium processor and its on-chip clockdriver circuit;

FIG. 3 is a diagram showing comparisons between a PLL of a computersystem and PLL of a communication system;

FIG. 4 is a diagram showing a basic configuration of a PLL circuit;

FIG. 5 is a block diagram showing an example of a phase-frequencydetector;

FIG. 6 is a state transition diagram of the phase-frequency detector;

FIG. 7 shows the operation waveforms of the phase-frequency detectorwhen a frequency error is negative;

FIG. 8 shows a charge pump circuit;

FIG. 9 shows a loop filter circuit;

FIG. 10 is a circuit diagram showing a passive lag filter;

FIG. 11 shows an example of a VCO circuit;

FIG. 12 shows an example of a jitter of a clock;

FIG. 13 is a diagram for explaining a method of measuring a jitter;

FIG. 14 is a diagram showing a spectrum of an output signal of an idealoscillator;

FIG. 15 is a diagram showing a variation of zero crossing caused by aphase noise;

FIG. 16 is a diagram showing a diffusion of a spectrum caused by a phasenoise;

FIG. 17 is a block diagram showing a VCO circuit in which a noise isadded to its input end;

FIG. 18 is a block diagram showing another VCO circuit equivalent to theVCO circuit in which a noise is added to its input end;

FIG. 19 is a block diagram showing a VCO circuit having an internalphase noise;

FIG. 20 is a block diagram showing a PLL circuit which simulates ajitter;

FIG. 21 is a diagram showing an ideal clock waveform;

FIG. 22 is a waveform diagram showing a clock waveform and itsHilbert-transformed result;

FIG. 23 is a diagram showing an example of a measured peak-to-peakjitter in time domain;

FIG. 24 is a typical model diagram showing a measuring system of apeak-to-peak jitter;

FIG. 25 is a diagram showing zero crossings and instantaneous periods;

FIG. 26 is a diagram showing a set of phase noises and a set of phasenoises which can be sampled by a zero crossing method;

FIG. 27 is a diagram showing the zero crossing in a complex plane;

FIG. 28 is a waveform diagram showing a measured example of an RMSjitter in frequency domain;

FIG. 29 is a typical model diagram showing a measuring system of an RMSjitter;

FIG. 30 is a diagram showing a clock as a random phase modulation signaland a clock as an analytic signal;

FIG. 31 is a diagram showing an oscillation waveform of a VCO as ananalytic signal;

FIG. 32 is a block diagram showing a first embodiment of an apparatusfor measuring a jitter according to the present invention;

FIG. 33 is a diagram showing a constant frequency signal for measuring ajitter;

FIG. 34 is a typical model diagram showing a jitter measuring systemwherein an apparatus for measuring a jitter according to the presentinvention is used;

FIG. 35 is a diagram for explaining the operation of a Hilbert pairgenerator;

FIG. 36 is a diagram for explaining a Hilbert-transforming and abandpass filtering;

FIG. 37 is a diagram for explaining the operation of an instantaneousphase estimator;

FIG. 38 is a diagram for explaining the operation of a linear phaseremover;

FIG. 39 is a waveform diagram showing a comparison between the zerocrossing method and the method of the present invention;

FIG. 40 is a block diagram showing an apparatus for measuring a jitterwherein a modulation sampling method is used;

FIG. 41 is a diagram showing differences between a sampling method inthe zero crossing method and a sampling method in the method of thepresent invention;

FIG. 42 is a diagram showing the fundamental wave spectrum and a clockwaveform;

FIG. 43 is a diagram showing a partial sum spectrum of up to 13th orderharmonics and a clock waveform;

FIG. 44 is a diagram showing a relative error included in a partial sumof up to a certain order of harmonics;

FIG. 45 is a diagram showing parameters of a MOSFET;

FIG. 46 is a block diagram showing a jitter-free PLL circuit;

FIG. 47 is a diagram showing waveforms at input and output of a VCO inthe jitter-free PLL circuit;

FIG. 48 is a diagram showing an output waveform and a waveform of aphase noise of a VCO in the jitter-free PLL circuit;

FIG. 49 is a diagram showing an instantaneous period and a waveform of aphase noise of the jitter-free PLL circuit;

FIG. 50 is a block diagram showing a jittery PLL circuit;

FIG. 51 is a diagram showing waveforms at input and output of a VCO inthe jittery PLL circuit;

FIG. 52 is a diagram showing an output waveform and a waveform of aphase noise of a VCO in the jittery PLL circuit;

FIG. 53 is a diagram showing an instantaneous period and a waveform of aphase noise of the jittery PLL circuit;

FIG. 54 is a diagram for explaining a method of estimating an RMS jitterby a spectrum method and a phase noise waveform estimating method;

FIG. 55 is a diagram for comparing estimated values of the RMS jitter;

FIG. 56 is a diagram for explaining a method of estimating apeak-to-peak-jitter by a zero crossing method and a phase noise waveformestimating method;

FIG. 57 is a diagram for comparing estimated values of the peak-to-peakjitter;

FIG. 58 is a waveform diagram showing a phase noise of a jitteryfrequency-divided clock;

FIG. 59 is a diagram for comparing estimated values of the RMS jitter ofa frequency-divided clock;

FIG. 60 is a diagram for comparing estimated values of the peak-to-peakjitter of a frequency-divided clock;

FIG. 61 is a waveform diagram showing a phase noise spectrum;

FIG. 62 is a waveform diagram showing an example of a Hilbert pair;

FIG. 63 is a waveform diagram showing another example of a Hilbert pair;

FIG. 64 is a waveform diagram for explaining a difference betweenpeak-to-peak jitters;

FIG. 65 is a diagram in which estimated values of the peak-to-peakjitter are plotted; and

FIG. 66 is a waveform diagram showing a VCO input and a PLL clock of adelay-fault free PLL circuit.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the study and development of a PLL circuit, a conventional method ofmeasuring a jitter is still utilized, and the compatibility between adata in a test stage and a data in a development stage is an importantproblem. Particularly, in order to make a design change in a shortperiod of time and/or in order to improve a process to realize animprovement of the production yield, a test method which can share thetest results is a key point. From this view point, the present inventionprovides a method and an apparatus which are reasonable as a clock testmethod.

In order to realize the compatibility with an RMS jitter, the shape of aphase noise power spectrum must be maintained in frequency domain. Thiscan be solved by using an analytic signal already discussed. Next, inorder to realize the compatibility with a peak-to-peak jittermeasurement, a method of maintaining the zero crossing of a waveform isrequired. Incidentally, as already shown clearly, the fundamental of aclock waveform maintains zero crossing information of the original clock(“theorem of zero crossing”). Therefore, for a measurement of apeak-to-peak jitter, it is sufficient to estimate a phase angleutilizing only the fundamental of the clock waveform. For example, theequation (2.5.2) or (3.23) corresponds to this fundamental wave.

From the equation (2.5.2) or (3.23), it can be interpreted that a phasenoise waveform Δφ(t) randomly changes a phase of a carrier wavecorresponding to the clock frequency. As a result of this random phasemodulation, a period of the carrier wave is fluctuated and hence ajitter is generated. An actually observable quantity is, as shown inFIG. 30(a), only a real part of the random phase modulation signal (forexample, refer to a reference literature c16). However, if an imaginarypart could also be observed simultaneously, a phase angle can easily beobtained. This concept corresponds to that the clock waveform isregarded as the aforementioned analytic signal. FIG. 30(b) illustrates ablock diagram when the clock waveform is regarded as an analytic signal.When the inside of the PLL circuit is considered, as shown in FIG. 31,an oscillation waveform of a voltage-controlled oscillator (VCO) couldbe handled as the analytic signal.

In this case, Δφ(t) randomly phase-modulates the clock signal.Therefore, it is an object of the present invention to provide a methodof deriving Δφ(t) from the clock waveform. FIG. 32 shows a block diagramof a first embodiment of an apparatus for measuring a jitter accordingto the present invention.

As already mentioned above, a reference clock signal which continues tostrictly maintain a constant period is applied to the PLL circuit undertest. FIG. 33 shows the reference clock signal. As a result, the PLLcircuit under test does not internally generate a phase error, and henceonly a random jitter caused by a VCO appears on the clock waveform. Anacquired clock waveform is transformed to an analytic signal, and itsinstantaneous phase is estimated to measure a jitter based on thedispersion from a linear phase. FIG. 34 shows a jitter test system towhich the present invention is applied.

Each block can also be realized by an analog signal processing. However,in the present invention, each block is practiced by a digital signalprocessing. Because, a digital signal processing is more flexible thanan analog signal processing, and its speed and accuracy can easily bechanged in accordance with the hardware cast. Conjecturing from thepresent inventors' experience in developing a noise analyzing apparatusfor a TV picture signal, the required number of bits in quantizing aclock waveform would be equal to or more than 10 bits.

Now, an algorithm for measuring a jitter used in the present inventionwill be described.

A Hilbert pair generator 11 shown in FIGS. 32 and 35 transforms a clockwaveform X_(C)(t) into an analytic signal Z_(C)(t). From the equation(3.6), the Hilbert-transformed result of X_(C)(t) is given by thefollowing equation (3.34).

{circumflex over (x)} _(c)(t)=H[x _(c)(t)]=A _(c) sin(2πf _(c) t+θ_(c)+Δφ(t))  (3.34)

If X_(C)(t) and {circumflex over (X)}_(C)(t) are assumed to be a realpart and an imaginary part of a complex number, respectively, ananalytic signal is given by the following equation (3.35).

z _(c)(t)=x_(c)(t)+j{circumflex over (x)} _(c)(t)=A _(c) cos(2πf _(c)t+θ _(c)+Δφ(t))+jA _(c) sin(2πf _(c) t+θ _(c)+Δφ(t))  (3.35)

The following Algorithm 3 is a computation or calculation procedureutilizing “theorem of zero crossing (the fundamental of a waveform holdsthe zero crossing information of the original waveform)”. That is, theAlgorithm 3 is the calculation procedure utilizing this demonstration.In other words, the Algorithm 3 transforms only the fundamental of aclock waveform into an analytic signal. FIG. 36(a) shows the originalclock waveform which has a shape close to a square wave. FIG. 36(b)shows a two-sided power spectrum which is the Fourier transformed resultof the clock waveform. Then the negative frequency components are cut.At the same time, as shown in FIG. 36(c), only the fundamental of theclock waveform is extracted by a bandpass filtering. That is, in thisstep, Hilbert-transform and the bandpass filtering are simultaneouslyperformed. When the spectrum shown in FIG. 36(c) isinverse-Fourier-transformed, an analytic signal is obtained. Since onlythe frequency components in the proximity of the fundamental areextracted by the bandpass filtering, the analytic signal shown in FIG.36(d) corresponds to the fundamental of the clock waveform, and X_(C)(t)indicated by a solid line is a sum of sine waves.

Algorithm 3 (Procedure to transform a real waveform into an analyticsignal of the fundamental thereof:

1. By using fast Fourier transform, X_(C)(t) is transformed intofrequency domain;

2. Negative frequency components are cut to zero. Frequency componentsin the proximity of the clock frequency are extracted by a bandpassfiltering, and other frequency components are cut to zero;

3. The spectrum is transformed into time domain using inverse fastFourier transform.

An instantaneous phase estimator 12 estimates an instantaneous phaseΘ(t); of X_(C)(t) using Z_(C)(t). That is, the following equation(3.36.1) is obtained.

Θ(t)=[2πf _(c) t+θ _(c)+Δφ(t)]mod 2π  (3.36.1)

Next, the instantaneous phase estimator 12 applies the aforementionedphase unwrap method to Θ(t). As a result, the following equation(3.36.2) is obtained, where θ(t) is a continuous phase.

θ(t)=2πf _(c) t+θ _(c)+Δφ(t)  (3.36.2)

FIGS. 37(b) and 37(c) show an instantaneous phase and an unwrappedphase, respectively. In addition, a linear phase remover 13 estimates,using the aforementioned linear trend estimating method, a linear phase[2πf_(C)(t)+θ_(C)] based on θ(t). Next, if the linear phase is removedfrom θ(t), a varying term Δφ(t) of the instantaneous phase, namely, aphase noise waveform expressed by the following equation (3.36.3) can beobtained.

θ(t)=Δφ(t)  (3.36.3)

FIG. 37(b) shows Δφ(t). The jitter measuring algorithm used in thepresent invention can estimate concurrently a peak-to-peak jitter J_(PP)and an RMS jitter J_(RMS) from Δφ(t). That is, the following equations(3.37) and (3.38) can be obtained. $\begin{matrix}{J_{PP} = {{\max\limits_{k}\left( {\Delta \quad \varphi \quad (k)} \right)} - {\min\limits_{l}\left( {\Delta \quad {\varphi (l)}} \right)}}} & \text{(3.37)} \\{J_{RMS} = \sqrt{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\Delta \quad {\varphi^{2}(k)}}}}} & \text{(3.38)}\end{matrix}$

Hereinafter, the method according to the present invention is alsoreferred to as Δφ(t) method.

Next, the method according to the present invention will be logicallycompared with the zero crossing method.

First, when only a rise edge of a signal (equal to the zero crossing) issampled, it will be proven that the Δφ(t) method is equivalent to thezero crossing method. Now, when the period of the zero crossing isexpressed as T_(ZERO), a clock waveform X_(C)(t) is expressed by thefollowing equation (3.39). $\begin{matrix}{{x_{c}(t)} = {A_{c}{\sin \left( {\frac{2\quad \pi}{T_{ZERO}}t} \right)}}} & (3.39)\end{matrix}$

Using the equation (3.35), an analytic signal expressed by the followingequation (3.40) is obtained. $\begin{matrix}\begin{matrix}{{z_{c}(t)} = {{x_{c}(t)} + {j{{\hat{x}}_{c}(t)}}}} \\{= {{A_{c}{\sin \left( {\frac{2\quad \pi}{T_{ZERO}}t} \right)}} - {j\quad A_{c}{\cos \left( {\frac{2\quad \pi}{T_{ZERO}}t} \right)}}}}\end{matrix} & \text{(3.40)}\end{matrix}$

From the equation (3.10.3), an instantaneous frequency of Z_(C)(t) isgiven by the following equation (3.41). $\begin{matrix}{{f(t)} = {\frac{\omega (t)}{2\quad \pi} = {\frac{{\Theta (t)}}{t} = \frac{{{x_{c}(t)}{{\hat{x}}_{c}^{\prime}(t)}} - {{{\hat{x}}_{c}(t)}{x_{c}^{\prime}(t)}}}{{x_{c}^{2}(t)} + {{\hat{x}}_{c}^{2}(t)}}}}} & (3.41)\end{matrix}$

Accordingly, f(t) can be expressed as follows. $\begin{matrix}{{f(t)} = \frac{1}{T_{ZERO}}} & \text{(3.42)}\end{matrix}$

That is, when only a rise edge of a signal is sampled, it has beenproven that the Δφ(t) method is equivalent to the zero crossing method.

In the zero crossing method, a time resolution of the period measurementcannot arbitrarily be selected. The time resolution of this method isdetermined by the zero crossing of the signal under measurement. On theother hand, in the Δφ(t) method, both time resolution and phaseresolution can be improved by increasing the number of samples perperiod. FIG. 39 shows a comparison between the data of the conventionalzero crossing method and the data of the Δφ(t) method. It cab be seenthat the time resolution on the time axis and the phase resolution onthe longitudinal axis have been improved.

Here, let's compare an upper limit of the sampling interval of the Δφ(t)method with that of the zero crossing method. The upper limit of thesampling interval of the Δφ(t) method can be derived from the conditionsdescribed above. That is, in order to uniquely perform a phase unwrap, aphase difference between adjacent analytic signals Z_(C)(t) must besmaller than π. In order for Z_(C)(t) to satisfy this condition, atleast two samples must be sampled in equal interval within a period. Forexample, since the frequency of X_(C)(t) given by the equation (3.23) isf_(C), the upper limit of the sampling interval is ½f_(C). On the otherhand, the upper limit of the equivalent sampling interval of the zerocrossing method is 1/f_(C).

Next, a sampling method using a quadrature modulation will be described.The clock frequency of a microcomputer has been being shifted to ahigher frequency in the rate of 2.5 times per 5 years. Therefore, unlessthe method of measuring a jitter is scalable with respect to a measuringtime resolution, a clock jitter of a microcomputer cannot be measured. Amethod of making the method of measuring a jitter scalable is thequadrature modulation. As can be seen from FIGS. 28 and 16, in a jitteryclock waveform, a phase noise spectrum is diffused from the clockfrequency as a central frequency. That is, a jittery clock waveform is aband-limited signal. For this reason, there is a possibility to decreasethe lower limit of the sampling frequency by combining the quadraturemodulation with a lowpass filter.

FIG. 40(a) is a block diagram showing a phase estimator for estimatingΔφ(t) of a clock waveform using a quadrature modulation system. InputtedX_(C)(t) is multiplied by the following equation (3.43) in a complexmixer.

cos(2π(f _(c) +Δf)t+θ)+j sin(2π(f _(c) +Δf)t+θ)  (3.43)

A complex output of the lowpass filter is given by the followingequation (3.44).

A _(c)/2[cos(2πΔft+(θ−θ_(c))−Δφ(t))+jsin(2πΔft+(θ−θ_(c))−Δφ(t))]  (3.44)

That is, the X_(C)(t) is transformed to an analytic signal Z_(C)(t) bythe quadrature modulation and the lowpass filter, and the frequency isdecreased to Δf. Thereafter, the analog signal is converted to a digitalsignal, and an instantaneous phase of the X_(C)(t) is estimated by aninstantaneous phase estimator so that an estimated instantaneous phaseΘ(t) expressed by the following equation (3.45) can be obtained.

Θ(t)=[2πΔft+(θ−θ_(c))−Δφ(t)]mod 2π  (3.45)

Similarly to the previous example, a phase unwrap is applied to the Θ(t)and a linear phase is removed by a linear phase remover so that thefollowing equation (3.46) can be obtained.

Θ(t)=−Δφ(t)  (3.46)

As mentioned above, it has been proven that the lower limit of thesampling frequency of the Δφ(t) method can be reduced from 2f_(C) to2(Δf) by combining the quadrature modulation with a lowpass filter.Similarly, the lower limit of the equivalent sampling frequency of thezero crossing method can also be reduced from f_(C) to Δf. A similareffect can also be obtained by combining a heterodyne system shown inFIG. 40(b) with a lowpass filter.

Finally, measuring time periods T_(meas) of the Δφ(t) method and thezero crossing method are derived. The T_(meas,ZERO) of the zero crossingmethod is given, by the following equation (3.47.1), as a time periodrequired for collecting the Δφ(t) of N points corresponding to the lowerlimit of an equivalent sampling frequency Δf. $\begin{matrix}{T_{{meas},{ZERO}} \geq \frac{N}{\Delta \quad f}} & \text{(3.47.1)}\end{matrix}$

On the other hand, regarding the Δφ(t) method, a case of K times of thenumber of samples per period will be discussed. Therefore, a time periodrequired for sampling, in the Δφ(t) method, N points of the Δφ(t) with afrequency 2K(Δf) which is K times as high as the frequency of the lowerlimit of sampling frequency is given by the following equation (3.47.2).$\begin{matrix}{T_{{meas},{\Delta \quad \varphi}} = {\frac{1}{2K}\left( \frac{N}{\Delta \quad f} \right)}} & \text{(3.47.2)}\end{matrix}$

That is, the Δφ(t) method can measure the Δφ(t) 2K times faster than thezero crossing method. In addition, in the Δφ(t) method, it can beunderstood that the measuring time resolution can be changed to bescalable by adjusting the K. On the contrary, the time resolution of thezero crossing method has been determined by Δf. FIG. 41 shows acomparison between the Δφ(t) method and the zero crossing method.

Next, a method of estimating a power spectrum density function of aphase noise waveform Δφ(t) will be explained. In the aforementionedAlgorithm 3, since only the fundamental is extracted by a bandpassfiltering, there is a drawback that a frequency range by which aspectrum distribution of the Δφ(t) can be observed is limited. Since anAlgorithm 4 described below is aimed to observe a spectrum distributionof the Δφ(t), a bandpass filtering is not used therein. Conversely, theAlgorithm 4 described below cannot be used for observing the Δφ(t).

When an analytic signal Z_(C)(t) is estimated, fast Fourier transform isused. In this case, X_(C)(t)W(t) (a waveform obtained by multiplying theX_(C)(t) by a window function W(t)) is fast-Fourier-transformed.Generally, an amplitude of the W(t) has a value close to zero at theproximity of the first time point and the last time point (for example,refer to a reference literature c17). For this reason, the waveformX_(C)(t)W(t) calculated by Inverse Fourier transform has a large errorat the proximity of the first time point and the last time point, andhence the X_(C)(t)W(t) cannot be used as a data. Also in the estimationof a Z_(C)(t), X_(C)(t)W(t) corresponding to the central portion, i.e.,approximately 50% of the window function is multiplied by a reciprocalof the window function, 1/W(t) to estimate the Z_(C)(t), and the valuesof the both ends of X_(C)(t)W(t) are obliged to be discarded.

In this method, only z_(C)(t) of 512 points can be estimated fromX_(C)(t) of 1024 points. In this case, it is assumed that X_(C)(t) isrecorded in a waveform recording buffer. In order to increase the numberof samples for Z_(C)(t), it is necessary to segment the waveformrecording buffer such that the waveform partially overlaps with thepreceding waveform, to calculate Z_(C)(t) corresponding to each timeinterval, and finally to compose each Z_(C)(t) to obtain the entirecomposite Z_(C)(t).

When Z_(C)(t) is estimated, a window function which applies only theminimum modulation to an amplitude of a X_(C)(t) should be used. Thewindow function which can satisfy this condition is Hanning (a referenceliterature c17). This has only the minimum, i.e., 1, spectrum at each ofthe upper sideband and the lower sideband. In this case, approximately25% of a waveform is overlapped.

Algorithm 4 (Procedure for estimating a spectrum of an analytic signal):

1. The X_(C)(t) is taken out from the starting address of the waveformrecording buffer;

2. The X_(C)(t) is multiplied by a window function W(t);

3. The product X_(C)(t)W(t) is transformed into frequency domain usingfast Fourier transform;

4. Only the negative frequency components are cut to zero;

5. The spectrum is transformed into time domain using inverse Fouriertransform to obtain Z_(C)(t)W(t);

6. The Z_(C)(t)W(t) is multiplied by a reciprocal of the window functionto obtain Z_(C)(t);

7. The X_(C)(t) is taken out from the waveform recording buffer. In thiscase, the X_(C)(t) is taken out such that the two adjacent X_(C)(t) areoverlapped with each other by approximately 25%; and

8. The above steps 2-7 are repeated until the entire Z_(C)(t) isobtained.

Next, there will be described a specific example in which theeffectiveness of the aforementioned method of measuring a jitter isverified through a simulation.

Relationship Between the Zero Crossing of a Clock Waveform and theFundamental of the Clock Waveform.

It will be verified using the ideal clock waveform shown in FIG. 21 that“the zero crossing of the fundamental of a waveform holds the zerocrossing information of the original waveform (theorem of zerocrossing)”. That is, a clock waveform is Fourier-transformed, thefundamental frequency component is left, and the frequency components ofthe second and higher harmonics are cut to zero. This spectrum isinverse-Fourier-transformed to obtain the restored waveform in timedomain. The period is estimated from the zero crossing of this waveform.FIG. 42(a) shows a spectrum from which the harmonics have been removed.FIG. 42(b) shows the restored waveform with the original clock waveformoverlaid thereon. Similarly, a partial sum spectrum of up to 13th orderharmonics and the restored waveform are plotted in FIGS. 43(a) and 43(b)respectively. Comparing each restored waveform with the original clockwaveform, it is seen that the zero crossing is a fixed point. That is, atime point of the zero crossing is constant regardless of the number oforders of harmonics used in the partial sum.

A relative error between “a period of the original clock waveform” and“a period estimated from a restored waveform” is obtained for each orderof harmonics by incrementing the number of harmonics from 1 to 13. FIG.44(a) shows the relative error values of the period. An error of theestimated period does not depend on the number of orders of harmonics.As a result, it has been verified that “the zero crossing of thefundamental gives a good approximation to the zero crossing of theoriginal signal”. The relative errors of root-mean-square values of awaveform are also given for a comparison purpose. FIG. 44(b) shows therelative errors of the root-mean-square values estimated from a restoredwaveform against the root-mean-square values of the original clockwaveform. It is seen in the root-mean-square case that the relativeerror is not decreased unless the partial sum includes higher orderharmonics.

Summarizing the above results, it could be understood that “if only thefundamental of a clock waveform can be extracted, an instantaneousperiod can be estimated from the zero crossing of the original clockwaveform. In this case, even if more harmonics are added in theestimation, the estimation accuracy is not improved”. That is, “theoremof zero crossing” has been proven.

Next, a case in which the method of measuring a jitter (Δφ(t) method)according to the present invention is applied to a jitter-free PLLcircuit will be explained. As a PLL circuit, the PLL circuit disclosedin the explanation of the prior art is used. A PLL shown in FIG. 46 iscomposed by 0.6 μm CMOSs and is operated by a power supply of 5 V toobtain various waveforms in a SPICE simulation. FIG. 45 shows parametersof a MOSFET. An oscillation frequency of the VCO is 128 MHz. A frequencydivider divides an oscillation waveform of the VCO to convert it into aPLL clock having 32 MHz frequency. The time resolution of the SPICEsimulation waveform is 50 psec. Then, a phase noise waveform Δφ(t) iscalculated from the simulation waveform. The estimation of the Δφ(t) issimulated using Matlab.

FIG. 47(a) shows an input waveform to the VCO. FIG. 47(b) shows anoscillation waveform of the VCO. FIG. 48(a) shows an output powerspectrum of the VCO. The oscillation waveform of the VCO of 8092 pointsis multiplied by “minimum 4 term window function” (for example, refer toa reference literature c18), and a power spectrum density function isestimated using fast Fourier transform. FIG. 48(b) shows the powerspectrum density function of the Δφ(t) estimated using the Algorithm 4.The condition of the fast Fourier transform operation is the same asthat used for obtaining the output power spectrum density function ofthe VCO. Comparing FIG. 48(a) with FIG. 48(b), in the power spectrum ofthe Δφ(t), it is seen that the spectrum of the oscillation frequency ofthe VCO of 128 MHz is attenuated by approximately 120 dB. The powerspectrum density function of the Δφ(t) has higher levels at lowerfrequencies due to an influence of a 1/f noise.

FIGS. 49(a) and 49(b) are diagrams for comparing the conventional zerocrossing method with the method according to the present invention. FIG.49(a) shows a result of the instantaneous period measurement of anoscillation waveform of the VCO measured by the zero crossing method.FIG. 49(b) shows the Δφ(t) estimated using the method of Algorithm 3according to the present invention. A spectrum of a frequency range (10MHz-200 MHz) in which the second order harmonic is not included wasextracted by a bandpass filter and the Δφ(t) was obtained by inversefast Fourier transform. It can be confirmed, from the fact that theinstantaneous period and the Δφ(t) do not indicate a noise, that thisPLL circuit does not actually have a jitter.

From FIG. 47(a), it can be seen that a frequency-up pulse is applied tothe VCO at the time point of approximately 1127 nsec. Two frequency-downpulses are applied to the VCO at the time points of approximately 908nsec and 1314 nsec, respectively. This is based on the performance ofthe PLL circuit used in the simulation. Viewing the Δφ(t) shown in FIG.49(b), a phase change due to the influence of the frequency-up pulseappears at the time point of approximately 1140 nsec. Phase changes dueto the influences of the two frequency-down pulses appear at the timepoints of approximately 920 nsec and 1325 nsec, respectively. These aredeterministic data. On the other hand, in the instantaneous period ofFIG. 49(a), a phase change due to the influence of the frequency-uppulse appears at the time point of approximately 1130 nsec. A phasechange due to the influence of the frequency-down pulses appears only atthe time point of approximately 910 nsec. An influence of afrequency-down pulse at approximately 1314 does not appear in the changeof the instantaneous period.

Summarizing the above results, in the Δφ(t) method according to thepresent invention, it can be observed that when a phase noise is notpresent, the oscillation state changes in accordance with a frequency-uppulse or a frequency-down pulse. The Δφ(t) method has a higherresolution compared with the conventional zero crossing method. Thepower spectrum density function of the Δφ(t) is influenced little by thespectrum of the oscillation frequency of the VCO.

Next, a case in which the aforementioned method of measuring a jitter(Δφ(t) method) according to the present invention is applied to ajittery PLL circuit will be explained. In addition, the method of thepresent invention will be compared with the instantaneous periodestimation using the zero crossing method to verify that the method ofmeasuring a jitter according to the present invention is effective for aphase noise estimation.

As already mentioned above, a jitter can be simulated by applying anadditive noise to the VCO to randomly modulate the phase of theoscillation waveform of the VCO. In the present invention, the jitter ofthe PLL circuit was simulated by applying an additive noise to an inputend of the VCO oscillation circuit. A Gaussian noise was generated usingthe Matlab's function randn0. Further, based on SPICE simulation, aGaussian noise was applied to an input end of the VCO of the PLL circuitshown in FIG. 50.

FIG. 51 shows an input waveform to the VCO when 3σ of the Gaussian noiseis 0.05 V. FIG. 51(b) shows an oscillation waveform of the VCO.Comparing FIG. 47(a) with FIG. 49(a), it is seen that the number offrequency-up pulses is increased from 1 to 4 and the number offrequency-down pulses is also increased from 2 to 3 due to the jitter.FIG. 52(a) shows the output power spectrum of the VCO. The noisespectrum has been increased. FIG. 52(b) shows a power spectrum densityfunction of the Δφ(t). Comparing FIG. 48(b) with FIG. 52(b), it is seenthat the power of the Δφ(t) has been increased. The power spectrumdensity function of the Δφ(t) has higher levels at lower frequencies.

FIGS. 53(a) and 53(b) are diagrams for comparing the conventional zerocrossing method with the method of measuring a jitter according to thepresent invention. FIG. 53(a) shows a result of the instantaneous periodmeasurement of an oscillation waveform of the VCO measured by the zerocrossing method. FIG. 53(b) shows the Δφ(t) estimated using the methodof measuring a jitter according to the present invention. Comparing FIG.53 with FIG. 49, it is seen that the corresponding waveform change issignificantly different from one another. That is, when no jitter ispresent, an instantaneous period and/or the Δφ(t) shows low frequencycomponents. On the other hand, when a jitter is present, aninstantaneous period and/or the Δφ(t) shows relatively high frequencycomponents. This means that the instantaneous period and/or the Δφ(t)shown in FIG. 53 corresponds to a phase noise. Further, if FIG. 53(a) iscarefully compared with FIG. 53(b), the following can be seen. (i) Theinstantaneous period and the Δφ(t) are slightly similar to each other.However, (ii) the time resolution and the phase (period) resolution ofthe Δφ(t) are higher than those of the instantaneous period.

Summarizing the above results, the method of measuring a jitter (theΔφ(t) method) according to the present invention can estimate a phasenoise with a high time resolution and a high phase resolution. Ofcourse, the zero crossing method can also estimate a phase noise in theform of an instantaneous period. However, there is a disadvantage in thezero crossing method that the time resolution and the period estimatingresolution are limited to the zero crossings.

Next, the conventional method of estimating a jitter will be comparedwith the method of measuring a jitter (the Δφ(t) method) according tothe present invention. However, with respect to the estimation of an RMSjitter, the Δφ(t) method will be compared with the spectrum method, andwith respect to the estimation of a peak-to-peak jitter, the Δφ(t)method will be compared with the zero crossing method.

FIG. 54 shows conditions for comparing the estimated values of the RMSjitter. The power spectrum density function of the Δφ(t) estimated usingthe aforementioned Algorithm 4 is used as a spectrum of the conventionalmethod. In the spectrum method, a sum of phase noise power spectrum inthe frequency range (10 MHz-200 MHz) in which the second order harmonicis not included is obtained and an RMS jitter J_(RMS) is estimated usingthe equation (3.33). The portion painted out in black in FIG. 54(a) isthe spectrum corresponding to the frequency range. On the other hand, inthe Δφ(t) method, an RMS jitter J_(RMS) is estimated using theaforementioned Algorithm 3 and the equation (3.38). This corresponds toa root-mean-square value of the phase noise waveform Δφ(t). The 3σ of aGaussian noise is applied to an input end of the VCO of the PLL circuitshown in FIG. 50 by changing its value from 0 to 0.5 V to estimate anRMS jitter value of an oscillation waveform of the VCO. As shown in FIG.55, the Δφ(t) method and the spectrum method provide the estimatedvalues substantially compatible with each other, respectively.

FIGS. 56(a) and 56(b) are diagrams for comparing estimated values of thepeak-to-peak jitter. Triangular marks indicate respective peak values.The positions of the triangular marks are different between the Δφ(t)method and the spectrum method. This means that a peak-to-peak jitter isnot necessarily be generated at the zero crossings. As shown in FIG. 57,the Δφ(t) method and the spectrum method provide the estimated valuescompatible with each other, respectively.

Summarizing the above results, the Δφ(t) method according to the presentinvention provides, in the estimation of an RMS jitter, estimated valuescompatible with those of the conventional spectrum method. Also in theestimation of a peak-to-peak jitter, the Δφ(t) method provides estimatedvalues compatible with those of the zero crossing method.

Next, the performance of the conventional method of estimating a jitterand the performance of the Δφ(t) method according to the presentinvention will be compared using a PLL clock which is frequency-dividedinto ¼ frequency. As the subject PLL circuit, the PLL circuit shown inFIG. 50 is used. The frequency divider of this circuit frequency-dividesan oscillation waveform of the VCO into ¼ frequency to convert theoscillation frequency to a PLL clock having 32 MHz frequency. FIG. 66(b)shows the waveform of the PLL clock. In addition, in order to comparewith the results of the aforementioned examples, the 3σ of an additiveGaussian noise is determined to be 0.05 V.

Assuming that the period of the oscillation waveform of the VCO isτ_(VCO), the period of the frequency-divided-by-four PLL clock τ_(PLL)is given by the following equation (3.48). $\begin{matrix}{\tau_{PLL} = {4\left( {\tau_{VCO} + \frac{\sum\limits_{j}^{4}ɛ_{i}}{4}} \right)}} & (3.48)\end{matrix}$

In this case, ε_(i) represents a time fluctuation of a rise edge. Fromthe equation (3.48), it can be understood that the frequency-divisionprovides an effect to reduce the jitter of the clock.

FIGS. 58(a) and 58(b) are diagrams for comparing the zero crossingmethod with the Δφ(t) method according to the present invention. FIG.58(a) shows a result of the measurement of an instantaneous period ofthe PLL clock in the zero crossing method. FIG. 58(b) shows the Δφ(t)estimated using the aforementioned Algorithm 3 of the Δφ(t) methodaccording to the present invention. A spectrum of a frequency range (20MHz-59 MHz) in which the second order harmonic is not included isextracted by a bandpass filter and the Δφ(t) is obtained by inverse fastFourier transform. It is seen that the Δφ(t) of the PLL clock issignificantly different from the Δφ(t) obtained from the oscillationwaveform of the VCQ shown in FIG. 58(b). The Δφ(t) of the PLL clockemphasizes phase discontinuity points. This is due to the frequencydivision. Because, the phase discontinuity points are in equal intervalsand are corresponding to the regular frequency-division edges.

FIG. 59 shows a diagram for comparing the estimated values of the RMSjitter. In the spectrum method, (i) the Δφ(t) was estimated from the PLLclock using the Algorithm 4 of the Δφ(t) method according to the presentinvention; (ii) the Δφ(t) of 8092 points was multiplied by “the minimum4 term window function” (for example, refer to a reference literaturec18) and the power spectrum density function was estimated by fastFourier transform. Next, (iii) in the spectrum method, a sum of phasenoise power spectrum in the frequency range (20 MHz-59 MHz) in which thesecond order harmonic is not included was obtained and an RMS jitterJ_(RMS) was estimated using the equation (3.33). On the other hand, inthe Δφ(t) method according to the present invention, an RMS jitterJ_(RMS) was estimated using the Algorithm 3 and the equation (3.38). Asshown in FIG. 59, the Δφ(t) method and the spectrum method provideestimated values substantially compatible with each other. However, inthe proximity of 0.05 V of 3σ of an additive Gaussian noise, the RMSjitter J_(RMS) estimated by the Δφ(t) method is larger than the RMSjitter of the spectrum method. The reason for this will be explainedtogether with the test result of the peak-to-peak jitter J_(PP).Comparing FIG. 59 with FIG. 55, it is seen that the frequency-divisionto ¼ frequency in this specific example makes J_(RMS) {fraction(1/3.7)}.

FIG. 60 shows a diagram for comparing the estimated values of thepeak-to-peak jitter. The Δφ(t) method and the zero crossing methodprovide estimated values substantially compatible with each other.However, in the proximity of 0.05 V of 3σ of an additive Gaussian noise,the peak-to-peak jitter J_(pp) estimated by the Δφ(t) method is largerthan the peak-to-peak jitter of the zero crossing method. Next, thereason for this will be explained.

FIG. 61 (a) shows a phase noise power spectrum when the 3σ is 0.05 V(substantially same estimated value as in the zero crossing method). Acursor in the figure indicates an upper limit frequency in estimatingthe Δφ(t). A weak phase modulation spectrum is recognized in theproximity of the cursor. FIG. 62 shows an analytic signal Z_(C)(t) inthis case. It is seen that the analytic signal Z_(C)(t) has become acomplex sine wave due to the weak phase modulation spectrum. For thisreason, the instantaneous phase smoothly changes.

FIG. 61(b) shows a phase noise power spectrum when the 3σ is 0.10 V(larger estimated value than that in the zero crossing method). Thisphase noise power spectrum shows a shape of typical 1/f noise. Thefundamental frequency of this 1/f noise is not the PLL clock frequency32 MHz. However, the Z_(C)(t) of the 1/f noise is given by the Hilbertpair of a square wave derived in the aforementioned example. Therefore,the Z_(C)(t) shown in FIG. 63 takes the same shape as the Hilbert pairshown in FIG. 30. Since the Z_(C)(t) has a complex shape, theinstantaneous phase largely changes. Therefore, the J_(PP) and theJ_(RMS) estimated by the Δφ(t) method take large values when the 3σ ofan additive Gaussian noise is close to 0.05 V. Comparing FIG. 60 withFIG. 57, it can be understood that the frequency-division to ¼ frequencyin this specific example makes the J_(PP)

Summarizing the above results, it has been verified that the Δφ(t)method can also estimate an RMS jitter and a peak-to-peak jitter of afrequency-divided clock. The estimated value is compatible with theconventional measuring method. However, when the phase noise powerspectrum takes a shape of 1/f noise, the Δφ(t) method indicates a largervalue than in the conventional measuring method.

As is apparent from the above discussion, the effectiveness of themethod of measuring a jitter (the Δφ(t) method) according to the presentinvention has been verified through a simulation. In addition, it hasbeen verified that the zero crossing of the original waveform can beestimated from the zero crossing of the fundamental. This has providedan important base that the Δφ(t) method can estimate a peak-to-peakjitter compatible with the zero crossing method. Because, if the Δφ(t)is estimated using the spectrum of an entire frequency range rather thanonly fundamental, a waveform shown in FIG. 56 is obtained. That is, aripple of higher frequencies is placed on top of the waveform. Further,as shown in FIG. 57, the compatibility with the zero crossing methodcannot be realized. Moreover, it has been verified that, when the Δφ(t)method is applied to a jittery PLL circuit, the Δφ(t) method iseffective for the phase noise estimation. In addition, it has been madeclear that the conventional method of measuring a jitter is compatiblewith the Δφ(t) method with respect to a peak-to-peak jitter and an RMSjitter. Further, it has been verified that a jitter of afrequency-divided clock can also be estimated with a compatibility.

As described above, the method of measuring a jitter of a clockaccording to the present invention comprises the signal processing stepsof: transforming a clock waveform X_(C)(t) into an analytic signal usingHilbert transform; and estimating a varying term Δφ(t) of aninstantaneous phase. This method has the following characteristics.

(i) The Δφ(t) method does not require a trigger signal. (ii) Apeak-to-peak jitter and an RMS jitter can simultaneously be estimated.(iii) A peak-to-peak jitter value estimated using the Δφ(t) method iscompatible with an estimated value in the conventional zero crossingmethod. (iv) An RMS jitter value estimated using the Δφ(t) method iscompatible with an estimated value in the conventional zero crossingmethod.

Further, the aforementioned reference literatures are as listed below.

[c1]: Alan V. Oppenheim, Alan S. Willsky and Ian T. Young, Signals andSystems, Prentice-Hall, Inc., 1983.

[c2]: Athanasios Papoulis, “Analysis for Analog and Digital Signals”,Gendai Kogakusha, 1982.

[c3]: Stefan L. Hahn, Hilbert Transforms in Signal Processing, ArtrchHouse Inc., 1996.

[c4]: J. Dugundji, “Envelopes and Pre-Envelopes of Real Waveforms,” IRETrans. Inform. Theory, vol. IT-4, pp. 53-57, 1958.

[c5]: Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time SignalProcessing, Prentice-Hall, Inc., 1989.

[c6]: Tristan Needham, Visual Complex Analysis, Oxford University Press,Inc., 1997.

[c7]: Donald G. Childers, David P. Skinner and Robert C. Kemerait, “TheCepstrum: A Guide to Processing,” Proc. IEEE, vol. 65, pp. 1428-1442,1977.

[c8]: Jose M. Tribolet, “A New Phase Unwrapping Algorithm,” IEEE Trans.Acoust., Speech, Signal Processing, vol. ASSP-25, pp. 170-177, 1977.

[c9]: Kuno P. Zimmermann, “On Frequency-Domain and Time-Domain PhaseUnwrapping,” Proc. IEEE, vol. 75, pp. 519-520, 1987.

[c10]: Julius S. Bendat and Allan G. Piersol, Random Data: Analysis andMeasurement Procedures, 2^(nd) ed., John Wiley & Sons, Inc., 1986.

[c11]: Shoichiro Nakamura, Applied Numerical Methods with Software,Prentice-Hall, Inc., 1991.

[c12]: David Chu, “Phase Digitizing Sharpens Timing Measurements,” IEEESpectrum, pp. 28-32, 1988.

[c13]: Lee D. Cosart, Luiz Peregrino and Atul Tambe, “Time DomainAnalysis and Its Practical Application to the Measurement of Phase Noiseand Jitter,” IEEE Trans. Instrum. Meas., vol. 46, pp. 1016-1019, 1997.

[c14]: Jacques Rutman, “Characterization of Phase and FrequencyInstabilities in Precision Frequency Sources: Fifteen Years ofProgress,” Proc. IEEE, vol. 66, pp. 1048-1075, 1977.

[c15]: Kamilo Feher, Telecommunications Measurements, Analysis, andInstrumentation, Prentice-Hall, Inc.,1987.

[c16]: Michel C. Jeruchim, Philip Balaban and K. Sam Shanmugan,Simulation of Communication Systems, Plenum Press, 1992.

[c17]: E. Oran Brigham, The Fast Fourier Transform, Prentice-Hall, Inc.,1974.

[c18]: Albert H. Nuttall, “Some Windows with Very Good SidelobeBehavior”, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29,pp. 84-91, 1981.

What is claimed is:
 1. An apparatus for measuring a jitter comprising:analytic signal transforming means for transforming a clock signal intoan analytic signal; and estimation means for estimating a varying termof an instantaneous phase of said analytic signal.
 2. The jittermeasuring apparatus according to claim 1, which further comprises jitterdetecting means for estimating a jitter of the clock signal from thevarying term of the instantaneous phase of said analytic signal.
 3. Amethod of measuring a jitter comprising the steps of: transforming aclock signal into an analytic signal; and estimating a varying term ofan instantaneous phase of said analytic signal.
 4. The jitter measuringmethod according to claim 3, which further comprises a jitter detectingstep of estimating a jitter of the clock signal in response to thevarying term of the instantaneous phase of said analytic signal.
 5. Anapparatus for measuring a jitter of a clock signal comprising: analyticsignal transforming means for converting only the fundamental frequencycomponent of a digital clock signal to be measured to a complex analyticsignal; estimation means for estimating a varying term of aninstantaneous phase of said analytic signal; and jitter detection meansfor estimating a jitter of the clock signal in response to the varyingterm.
 6. The jitter measuring apparatus according to claim 2 or 5,wherein said analytic signal transforming means comprises: means fortransforming the clock signal to a frequency domain signal; means forcutting-off negative frequency components from said frequency domainsignal thereby extracting positive frequency components of saidfrequency domain signal; and means for inverse-transforming an output ofsaid positive frequency components to a time domain signal.
 7. Thejitter measuring apparatus according to claim 6, wherein said jitterdetecting means comprises means for transforming the varying term to afrequency domain signal by using fast Fourier transform therebyobtaining a spectrum of the varying term.
 8. The jitter measuringapparatus according to claim 2 or 5, wherein said analytic signaltransforming means comprises: frequency domain transforming means fortransforming the clock signal to a frequency domain signal by using fastFourier transform; bandpass filtering means for cutting-off negativefrequency components from an output of said frequency domaintransforming means threreby extracting positive frequency components ofthe clock signal; and time domain transforming means forinverse-transforming an output of said bandpass filtering means to atime domain signal by using inverse fast Fourier transform.
 9. Thejitter measuring apparatus according to claim 8, wherein said jitterdetecting means comprises means for transforming the varying term to afrequency domain signal by using fast Fourier transform threrebyobtaining a spectrum of the varying term.
 10. The jitter measuringapparatus according to claim 2 or 5, wherein said analytic signaltransforming means comprises: means for transforming the clock signal toa frequency domain signal; filtering means for cutting-off negativefrequency components and extracting only frequency components in theproximity of a frequency of a fundamental of the clock signal from saidfrequency domain signal; and means for inverse-transforming an output ofsaid filtering means to a time domain signal.
 11. The jitter measuringapparatus according to claim 2 or 5, wherein said analytic signaltransforming means comprises: frequency domain transforming means fortransforming the clock signal to a frequency domain signal by using fastFourier transform; bandpass filtering means for cutting-off negativefrequency components and extracting only frequency components in theproximity of a frequency of a fundamental of the clock signal from anoutput of said frequency domain transforming means; and time domaintransforming means for inverse-transforming an output of said bandpassfiltering means to a time domain signal by using inverse fast Fouriertransform.
 12. The jitter measuring apparatus according to claim 2 or 5,wherein said estimation means comprises instantaneous phase estimationmeans for estimating an instantaneous phase of the analytic signal;conversion means for converting the instantaneous phase to a continuousphase; linear phase estimation means for estimating a linear phase fromthe continuous phase; and subtraction means for subtracting the linearphase from the continuous phase to thereby obtain the varying term ofthe instantaneous phase of said analytic signal.
 13. The jittermeasuring apparatus according to claim 2 or 5, which further comprises:an AD converter for converting an analog clock signal to be measured tosaid digital clock signal to be applicable to said analytic signaltransforming means.
 14. The jitter measuring apparatus according toclaim 2 or 5, wherein said analytic signal transforming means is meansfor performing both Hilbert-transform and band-pass filtering operationswherein the clock signal is band-pass filtered to extract only afrequency component of a fundamental of the clock signal whichcorresponds to a real part of the analytic signal, and the extractedoutput is Hilbert-transformed which corresponds to an imaginary part ofthe analytic signal.
 15. The jitter measuring apparatus according toclaim 2 or 5, wherein said jitter detecting means is peak-to-peakdetecting means for obtaining a difference between the maximum value andthe minimum value of the varying term of the instantaneous phase of saidanalytic signal as a peak-to-peak jitter.
 16. The jitter measuringapparatus according to claim 2 or 5, wherein said jitter detecting meansis root-mean-square detecting means for calculating a root-mean-squareof the varying term of the instantaneous phase of said analytic signalto thereby obtain a root-mean-square jitter.
 17. The jitter measuringapparatus according to claim 2 or 5, wherein said analytic signaltransforming means comprises: a first frequency mixer for multiplying ananalog clock signal to be measured by a sine wave signal; a secondfrequency mixer for multiplying the analog clock signal by a cosine wavesignal whose frequency is same as that of the sine wave signal; a firstlowpass filter to which an output of said first frequency mixer issupplied; a second lowpass filter to which an output of said secondfrequency mixer is supplied; a first AD converter for converting anoutput of said first lowpass filter to a digital signal; and a second ADconverter for converting an output of said second lowpass filter to adigital signal; whereby said analytic signal is composed of the digitaloutput signal of said first AD converter and the digital output signalof said second AD converter.
 18. The jitter measuring apparatusaccording to claim 2 or 5, which further comprises: a frequency mixermultiplying an analog clock signal to be measured by a cosine wavesignal; a lowpass filter to which an output of said frequency mixer issupplied; and an AD converter converting an output signal of the lowpassfilter to the digital clock signal to be measured which is applicable tosaid analytic signal transforming means.
 19. The jitter measuringapparatus according to claim 2 or 5, wherein said analytic signaltransforming means comprises: a buffer memory storing the clock signal;means for taking out in the sequential order a portion signal of theclock signal from said buffer memory in such a manner that the portionsignal of the clock signal taken out is partially overlapped with aprecedent portion signal of the clock signal which has been taken outjust before; means for multiplying each portion signal of the clocksignal by a window function; a frequency domain transformer transformingthe multiplied result of the multiplying means to a frequency domainsignal; a bandpass processor extracting only positive frequencycomponents of the clock signal by cutting-off negative frequencycomponents from the multiplied output of said frequency domaintransformer; a time domain transformer inverse-transforming the outputof said bandpass processor into a time domain signal; and means formultiplying the time domain signal transformed by a reciprocal of thewindow function to thereby obtain the analytic signal.
 20. The jittermeasuring apparatus according to claim 2 or 5, wherein the clock signalto be measured is an output of a PLL circuit.
 21. A method of measuringa jitter of a clock signal comprising the steps of transforming only thefundamental frequency component of a digital clock signal to be measuredto a complex analytic signal; estimating a varying term of aninstantaneous phase of said analytic signal; and detecting a jitter ofthe clock signal in response to the varying term.
 22. The jittermeasuring method according to claim 4 or 21, wherein said analyticsignal transforming step comprises: a frequency domain transforming stepof transforming the clock signal to a frequency domain signal; abandpass processing step of cutting-off negative frequency componentsfrom said frequency domain signal threreby extracting positive frequencycomponents of said frequency domain signal; and a time domaintransforming step of inverse-transforming an output of said bandpassprocessing step to a time domain signal.
 23. The jitter measuring methodaccording to claim 22, wherein said jitter detecting step comprises astep of transforming the varying term to a frequency domain signal byusing fast Fourier transform threreby obtaining a spectrum of thevarying term.
 24. The jitter measuring method according to claim 4 or21, wherein said analytic signal transforming means comprises: frequencydomain transforming step of transforming the clock signal to a frequencydomain signal by using fast Fourier transform; a bandpass filtering stepof cutting-off negative frequency components from an output of saidfrequency domain transforming means threreby extracting positivefrequency components of the clock signal; and a time domain transformingstep of inverse-transforming an output of said bandpass filtering stepto a time domain signal by using inverse fast Fourier transform.
 25. Thejitter measuring method according to claim 24, wherein said jitterdetecting step comprises a step of transforming the varying term to afrequency domain signal by using fast Fourier transform threrebyobtaining a spectrum of the varying term.
 26. The jitter measuringmethod according to claim 4 or 21, wherein said analytic signaltransforming step comprises; a frequency domain transforming step oftransforming the clock signal to a frequency domain signal; a bandpassprocessing step of cutting-off negative frequency components from thefrequency domain signal and extracting only frequency components in theproximity of a frequency of a fundamental of the clock signal; and atime domain transforming step of inverse-transforming an output of saidbandpass processing step to a time domain signal.
 27. The jittermeasuring method according to claim 4 or 21, wherein said analyticsignal transforming step comprises: a frequency domain transforming stepof transforming the clock signal to a frequency domain signal by usingfast Fourier transform; a bandpass filtering step of cutting-offnegative frequency components and extracting only frequency componentsin the proximity of a frequency of a fundamental of the clock signalfrom an output of said frequency domain transforming step; and a timedomain transforming step of inverse-transforming an output of saidbandpass filtering step to a time domain signal by using inverse fastFourier transform.
 28. The jitter measuring method according to claim 4or 21, wherein said estimating step comprises: an instantaneous phaseestimation step of estimating an instantaneous phase of the analyticsignal; a conversion step of converting the instantaneous phase to acontinuous phase; an linear phase estimation step of estimating a linearphase from the continuous phase; and a linear phase removing step ofsubtracting the linear phase from the continuous phase to thereby obtainthe varying term of the instantaneous phase of said analytic signal. 29.The jitter measuring method according to claim 4 or 21, which furthercomprises: an AD conversion step of converting an analog clock signal tobe measured to said digital clock signal which is to be applicable tosaid analytic signal transforming means.
 30. The jitter measuring methodaccording to claim 4 or 21, wherein said analytic signal transformingstep is a step of performing both Hilbert-transform and band-passfiltering operations wherein the clock signal is band-pass filtered tocut-out negative frequency component and to extract only a frequencycomponent of a fundamental of the clock signal which corresponds to areal part of the analytic signal, and the extracted output isHilbert-transformed which corresponds to an imaginary part of theanalytic signal.
 31. The jitter measuring method according to claim 4 or21, wherein said jitter detecting step is a peak-to-peak detecting stepof obtaining a difference between the maximum value and the minimumvalue of a phase noise waveform as a peak-to-peak jitter.
 32. The jittermeasuring method according to claim 4 or 21, wherein said jitterdetecting step is a root-mean-square detecting step of calculating aroot-mean-square of a phase noise waveform to obtain a root-mean-squarejitter.
 33. The jitter measuring method according to claim 4 or 21,wherein said analytic signal transforming step comprises: a firstfrequency mixing step of multiplying an analog clock signal to bemeasured by a sine wave signal; a second frequency mixing step ofmultiplying the analog clock signal by a cosine wave signal whosefrequency is same as that of the sine wave signal; a first filteringstep of applying a lowpass filtering process to an output signal fromsaid first frequency mixing step; a second filtering step of applying alowpass filtering process to an output signal from said second frequencymixing step; a first AD conversion step of converting an output fromsaid first filtering step to a digital signal; and a second ADconversion step of converting an output from said second filtering stepto a digital signal; whereby said analytic signal is composed of thedigital output signal of said first AD conversion step and the digitaloutput signal of said second AD conversion step.
 34. The jittermeasuring method according to claim 4 or 21, which further comprises: afrequency mixing step of multiplying an analog clock signal to bemeasured by a cosine wave signal; a lowpass filtering step of applying alowpass filtering process to the multiplied signal obtained from saidfrequency mixing step; an AD conversion step of converting a signalobtained from the lowpass filtering step to a digital signal which isapplicable to said analytic signal transforming step.
 35. The jittermeasuring method according to claim 4 or 21, wherein said analyticsignal transforming step comprises; a step of storing the clock signalin a buffer memory; a step of taking out in the sequential order aportion signal of the clock signal from said buffer memory such that theportion signal of the clock signal taken out is partially overlappedwith a precedent portion signal of the clock signal which has been takenout just before; a step of multiplying each portion signal of the clocksignal by a window function; a step of transforming each multipliedportion signal of the clock signal to a frequency domain signal; abandpass processing step of cutting-off negative frequency componentsfrom the frequency domain signal and of extracting only frequencycomponents in the proximity of a frequency of a fundamental wave of theclock signal; a step of inverse-transforming the bandpass-processedsignal to a time domain signal; and a step of multiplying the timedomain signal by a reciprocal of the window function to thereby obtainthe analytic signal.
 36. The jitter measuring method according to claim4 or 21, wherein the clock signal to be measured is an output of a PLLcircuit.